The specific claim is made in the Primary Chronicle, a twelfth-century history of the Kievan Rus. The Primary Chronicle entries for the years 986 and 987 say (1953 translation by Cross)

[986] Vladimir was visited by Bulgars of Mohammedan faith, who said, "Though you are a wise and prudent prince, you have no religion. Adopt our faith, and revere Mahomet."

Vladimir inquired what was the nature of their religion. They replied that they believed in God, and that Mahomet instructed them to practice circumcision, to eat no pork, to drink no wine, and, after death, promised them complete fulfillment of their carnal desires. "Mahomet," they asserted, "will give each man seventy fair women. He may choose one fair one, and upon that woman will Mahomet confer the charms of them all, and she shall be his wife. Mahomet promises that one may then satisfy every desire, but whoever is poor in this world will be no different in the next." They also spoke other false things which out of modesty may not be written down.

Vladimir listened to them, for he was fond of women and indulgence, regarding which he heard with pleasure. But circumcision and abstinence from pork and wine were disagreeable to him. "Drinking," said he, "is the joy of the Russes. We cannot exist without that pleasure."

Then came the Germans, asserting that they were come as emissaries of the Pope. They added, "Thus says the Pope: 'Your country is like our country, but your faith is not as ours. For our faith is the light. We worship God, who has made heaven and earth, the stars, the moon, and every creature, while your gods are only wood.'"

Vladimir inquired what their teaching was. They replied, "Fasting according to one's strength. But whatever one eats or drinks is all to the glory of God, as our teacher Paul has said." Then Vladimir answered, "Depart hence; our fathers accepted no such principle.''

The Jewish Khazars heard of these missions, and came themselves saying, "We have learned that Bulgars and Christians came hither to instruct you in their faiths. The Christians believe in him whom we crucified, but we believe in the one God of Abraham, Isaac, and Jacob.''

Then Vladimir inquired what their religion was. They replied that its tenets included circumcision, not eating pork or hare, and observing the Sabbath. The Prince then asked where their native land was, and they replied that it was in Jerusalem.

When Vladimir inquired where that was, they made answer, "God was angry at our forefathers, and scattered us among the gentiles on account of our sins. Our land was then given to the Christians.''

The Prince then demanded, "How can you hope to teach others while you yourselves are cast out and scattered abroad by the hand of God? If God loved you and your faith, you would not be thus dispersed in foreign lands. Do you expect us to accept that fate also?"

Then the Greeks sent to Vladimir a scholar... [This is followed by a long discussion of Christian theology.]

[987] Vladimir summoned together his boyars and the city elders, and said to them, "Behold, the Bulgars came before me urging me to accept their religion. Then came the Germans and praised their own faith; and after them came the Jews. Finally the Greeks appeared, criticizing all other faiths but commending their own, and they spoke at length, telling the history of the whole world from its beginning. Their words were artful, and it was wondrous to listen and pleasant to hear them. They preach the existence of another world. 'Whoever adopts our religion and then dies shall arise and live forever. But whosoever embraces another faith, shall be consumed with fire in the next world.' What is your opinion on this subject, and what do you answer?"

The boyars and the elders replied, "You know, oh Prince, that no man condemns his own possessions, but praises them instead. If you desire to make certain, you have servants at your disposal. Send them to inquire about the ritual of each and how he worships God."

Their counsel pleased the prince and all the people, so that they chose good and wise men to the number of ten, and directed them to go first among the Bulgars and inspect their faith. The emissaries went their way, and when they arrived at their destination they beheld the disgraceful actions of the Bulgars and their worship in the mosque; then they returned to their country. Vladimir then instructed them to go likewise among the Germans, and examine their faith, and finally to visit the Greeks. They thus went into Germany, and after viewing the German ceremonial, they proceeded to Tsargrad, where they appeared before the Emperor...

The envoys reported, "When we journeyed among the Bulgars, we beheld how they worship in their temple, called a mosque, while they stand ungirt. The Bulgar bows, sits down, looks hither and thither like one possessed, and there is no happiness among them, but instead only sorrow and a dreadful stench. Their religion is not good.

"Then we went among the Germans, and saw them performing many ceremonies in their temples; but we beheld no glory there.

"Then we went to Greece, and the Greeks led us to the edifices where they worship their God, and we knew not whether we were in heaven or on earth."

[And eventually Vladimir was baptized.]

Now, some context. The Primary Chronicle is really the only early source we have for this time period, but historians continue to debate just how reliable it is as a source. /u/hamiltonkg gives an example of a different episode in Rus' history WRT the Chronicle here.

But it's safe to say that details and embellishments as specific as Vladimir's investigation into the Abrahamic religions here are, at the very least, significantly exaggerated. (The story is also suspiciously similar to a story about the Khazar khan seeking out religions to convert to, only Judaism won out in that version.)

Now the question is, did Vladimir have an interest in Islam at all? If the Primary Chronicle were our only source, the question might be unanswerable. But the Islamic side actually somewhat corroborates this.

Around the same time as the Primary Chronicle, the Persian polymath Sharaf az-Zamān Ṭāhir Marwazī wrote a book titled the Nature of Animals. The first chapter discusses different varieties of humans, among them the Rus':

When they [the Rus'] entered the fold of Christianity, the faith blunted their swords, the door of their livelihood was closed to them, they returned to hardship and poverty, and their livelihood shrank. Then they desired to become Muslims, that it might be lawful for them to make raids and holy war, and so make a living by returning to their former practices. They therefore sent messengers to the ruler of Khwarazm, four kinsmen of their king; for they had an independent king called Vlādmīr, just as the king of the Turks is called khaqan and the king of the Bulghars yiltawar.

Their messengers came to Khwarazm and delivered their message, The Khwarazm-shah was delighted at their eagerness to become Muslims, and sent someone to them to teach them the religious laws of Islam. So they were converted.

The notion that the Rus' polity ever converted from Christianity to Islam should be dismissed, but the fact that Marwazī can name Vladimir specifically means the episode must have some backing in fact. It's very possible that these Rus' envoys who visited the Khwarazm court were the same people as the Rus' envoys Vladimir sent to the Bulghars, since the Bulghars received their training in the Islamic sciences from the cities of Khwarazm.

What, then, made Grand Prince Vladimir both 1)consider Islam as a religion to convert to, and 2)ultimately choose Byzantine Christianity over it?

Vladimir's accession to the throne was a fraught one. His mother was not his father Sviatoslav's most honored partner, and he had to treacherously kill his own half-brother Yaropolk to seize the throne. This meant the Grand Prince was particularly invested in religious legitimization, and at first he did this by emphasizing traditional pagan worship. The Primary Chronicle reports:

Vladimir then began to reign alone in Kiev, and he set up idols on the hills outside the castle with the hall: one of Perun, made of wood with a head of silver and a mustache of gold, and others of Khors, Dazh'bog, Stribog, Simar'gl, and Mokosh'.

But Vladimir was defeated by the Muslim Bulgars in the year 985. Historian Jonathan Shepherd notes that "the 'pantheon' of [pagan] gods was seen publicly to be failing to deliver," and Vladimir turned to the sure-and-tested legitimizing power of the Abrahamic religions.

This included Islam, which the Primary Chronicle mentions as the first religion to try and convert Vladimir. The fact that the Bulgars had apparently forced the Rus' to withdraw just the year before may not be coincidence. Did Vladimir's inquiry into the faiths begin by testing the strength of the Islamic God that had defeated him?

Ultimately, Vladimir chose Greek Christianity. Was this really because he wanted to keep drinking, as the Primary Chronicle says? Maybe, who knows. But what seems more likely to me is that Christianity was already more entrenched in 980s Rus' society than Islam was. There is far more archaeological and historical evidence of individual Christian converts before Vladimir's conversion, most notably the Prince's own grandmother St. Olga, than there are for Muslim converts. At the same time, Islam and Judaism were the religions of immediately adjacent enemies (the Bolgars and the Khazars), while Greek Christianity was associated more with the distant and opulent Byzantine Empire. Christianity was overall the safer option.

I've written at length about this before in this thread along with /u/facepoundr. The TL;DR is this:

Maybe but it's very unlikely. The more likely reason is that he accepted christianity as part of the bargain so that he could marry Princess Anna of Byzantium, sister to Emperor Basil II.