There's a few parts to this. Firstly, there's a sura of the Qur'an called Sura "ar-Rum", i.e. Sura of "Rome" or "the Roman Sura". We know from context that it is not referring to the city of Rome or to the Western Empire, but rather to the Greek speaking eastern empire, whom we typically refer to as the Byzantines. NB that the Byzantines themselves would have called themselves "Roman" so while this is somewhat confusing to us it would have been unambiguous at the time.

The text of this Sura includes a prediction the Byzantines would be defeated and then recover in their battles against the Persians. So from the Pickthall translation:

The Romans have been defeated

In the nearer land, and they, after their defeat will be victorious

Within ten years - Allah's is the command in the former case and in the latter - and in that day believers will rejoice

In Allah's help to victory. He helpeth to victory whom He will. He is the Mighty, the Merciful.

This is actually one of the only explicit references to contemporary events in the Qur'an, which generally speaking does not make specific references to contemporary global politics and those references that it does make are often ambiguous, despite the supposed clarity of later exegesis ascribing each reference to a point in the life of Muhammad.

So it is prophesying that the Byzantines, having suffered a major defeat at the hands of the Sassanians, would then gain victory.

So there is a Quranic understanding that the position of the Byzantines in global affairs is a matter of theological concern, but there's little explicitly about the Byzantines (Romans) beyond that except the more general references to the predicted triumphs of the believers.

Battles against the Romans (Byzantines) feature somewhat more prominently in the Hadith, the collected sayings and of the Prophet, for example here:

https://sunnah.com/muslim/54/50

and here: https://sunnah.com/muslim/54/46

But that's still basically pretty pedestrian stuff.

As far as I'm aware, though I've read much less of this literature, the vast bulk of what gets modern Salafi Jihadists excited about this stuff relates to the later, non-canonical apocalyptical literature that while sometimes attributed to Muhammad in the style of hadith is typically composed, at best, of "weak" hadith and from an academic perspective appears to be very closely related to mid-to-late 7th century campaigns against Byzantium and the assaults Constantinople. In terms of content this stuff is basically asserting that the Muslims will be victorious in Constantinople and then will march on the city of Rome and that this will be part of ushering in the apocalypse.

While such traditions have survived across the centuries they've never been regarded as being entirely canonical. Jihadists of an apocalyptic bent have latched on to them, particularly as they can read into the texts America or Russia or the Jews/Israel for Rome or whatever. These analogies are made pretty explicit in ISIS' materials and propaganda.

Source wise:

The appendices of Will McCant's ISIS Apocalypse include a number of these stories about the conquest victory and the end times. IIRC Robert Hoyland's In God's Path talks about the formation of the apocalyptical literature in the context of the campaigns against the Byzantines.

From my understanding the Islamic Ottomans knew they had conquered the Romans at the fall of Constantinople. Richard Fidler's history of Constantinople called "Ghost Empire" says "The long story of the Roman empire comes to a close with those two elegiac lines of Persian poetry uttered by Mehmed in the Great Palace."

The lines were: "The spider weaves the curtains in the Palace of the Caesars; The owl calls the watches in the towers of Afrasiab."

So at the fall of Constantinople in 1453 there is some evidence that the Muslim Turks understood the gravitas of Rome and what they had done to the empire.

Sultan Mehmed was also said to have adopted the title 'Kayser-i Rûm' which means 'Caesar of the Romans'. It also suggests that they themselves could not picture a world without a Roman presence and since they were the people to end the 2nd Rome, the mantle had been passed to the Islamic Turks to carry on Rome's legacy.

A Muslim Caesar didn't fly in Europe though, as a marriage between the Grand Prince of Moscow, Ivan III and the niece of the last emperor of Byzantium by the Pope at St. Peter's Basilica, in Europe's eyes, passed the legacy of Rome to Russia - hence the origin of the title 'Czar': 'Caesar'.