This is more or less correct but the problem is that the source of that story is an unreliable narrator: Napoléon himself! The main narrative about his alleged attempt at converting to Islam can be found in his memoir Campagnes d'Italie, d'Égypte et de Syrie. One can read it here (in French). Napoléon wrote it when exiled in Saint Helena, and the text was edited and published in 1870 with the rest of his correspondance (volume 29), half a century later.

Part1. The sources

Napoléon (then called Bonaparte) begins by claiming that "religious ideas were always prominent for the peoples of Egypt", and gives as an example of a successful conqueror Alexander the Great (who else!), who flattered the Egyptians by having a local priestess declare him the son of Jupiter. Napoléon cites Volney, the orientalist who explained Egypt to him and wrote that fighting Muslims was perhaps "an insurmountable obstacle". Napoléon notes that the French army, since its arrival in Egypt, had been "only tolerated" by the populations, who were grumbling openly against the "idolaters". So:

It was necessary to retreat or to conciliate religious ideas, to avoid the anathemas of the Prophet, not to allow oneself to be placed in the ranks of the enemies of Islamism; it was necessary to convince, to win over the muftis, the ulemas, the sheriffs, the imams, so that they would interpret the Koran in favour of the army.

The next step, Napoléon says, was to put the scholars of the Al-Azhar University on his side. He invites the heads (muftis) of the four schools of jurisprudence (Hanafi, Shafi'i, Maliki, Hanbali) to discuss with him. He sits down with them ("among them, on the same divan") and asks them to explain the Koran to him, and he praises the Prophet. Next thing you know, the muftis are fully convinced by the future Emperor, and they go back to the mosques ("respected by the French administration and even protected") where they tell the population that everything is going to be all right. Napoléon then says that the muftis had been deprived of their secular power by the Turks and by local Christians, and that he put them in charge again:

Everything was put back in order. The old custom was in all respects restored; this filled the Muslims with joy and inspired them with complete confidence.

By then, Napoléon was known as the "Sultan El-Kebir", the man who defeated the godless Mameluks thanks to the protection of the Prophet. In his memoirs, Napoléon refers to himself in the third person, and in this part of the text he switches between the (anachronistic) Napoléon and the Sultan El-Kebir.

The Sultan, "playing the card of Arab patriotism", tells his interlocutors that it is Cairo, not Constantinople, that is the centre of Islam, and that the Prophet would come to live in the Al-Azhar mosque if he ever came down to Earth. The "venerable old men" are now extatic and shout "Tayeb, tayeb! Ah, this is so true". They're completely in love with Napoléon! But the population and some imams still grumble about the unfaithful, and the Sultan El-Kebir complains "bitterly" about this to the sheiks: he wants a fatwa from Al-Azhar that forces the population to swear allegiance to him. The sheiks are a little befuddled by this, and this is when an ulema suggests that the Sultan (and his army?) convert to islam, and to this the sheiks all light up.

But Napoléon is unconvinced:

There are two great difficulties that stand in the way of me and my army becoming Muslims: the first is circumcision, the second is wine. My soldiers are used to it from childhood, I can never persuade them to give it up.

One sheik proposes to study the question (basically "let's make a committee") and this was a big success:

The self-esteem of all the Muslims was flattered, the joy was general.

Forty days later, the four muftis came back with a fawta that said that circumcision was "only recommended" and that drinking wine was possible for a Muslim, but that it was sinful and would deprive the person of his/her expected rewards.

Napoleon expressed his satisfaction at the solution of the first question; his joy seemed sincere [remember: he's talking about himself!]. All these old sheiks shared it.

But Napoléon was not happy with the wine part: he could not ask his soldiers to become Muslims if it made them "rebellious against the commands of heaven", so the fatwa went back to the drawing board. The muftis kept the "no circumcision" part, and asked Mecca for advice on the wine problem. A second fatwa was issued, which allowed those new Muslims to drink wine if they could dedicate one fifth of their income to almsgiving instead of the customary one tenth.

The victorious Napoléon concludes:

The sheiks, perfectly reassured, gave themselves entirely to the service of Sultan El-Kebir, and they understood that he needed at least a year to enlighten the minds and overcome the resistance. He had drawings, plans and budget estimates made for a mosque large enough to hold the whole army on the day it recognised the law of Muhammad. During this time, General Menou publicly embraced Islam. As a Muslim, he went to the mosque of Rosetta. He asked for no restrictions. This news filled the whole population of Egypt with joy, and left no doubt as to the sincerity of the hopes which they conceived. Everywhere the sheiks preached that Napoleon, being no infidel, loving the Koran, having a mission from the Prophet, was a true servant of the holy Kaaba. [...] Whenever the general-in-chief appeared in the city the faithful prostrated themselves; they behaved towards him as they were accustomed to do towards the sultan.

Indeed, a proclamation of the ulemas of Al-Azhar, dated from May or June 1799, is a panegyric of Napoléon (who cites it at length in his memoirs, volume 30), and ends with:

When the General-in-Chief arrived in Cairo, he informed the divan that he loved the Muslims, cherished the Prophet, instructed himself by reading the Koran every day. We know that he intends to build a mosque unrivalled in splendour and to embrace the faith of Muhammad.

So, according to him, Napoléon tried hard to get the religious authorities on his side, flattering them as much as possible, and ended up making vague promises ("a least a year" before he could convince his army to convert to Islam, "I'll build a mosque") in exchange for their collaboration and a nice proclamation. He never mentions the conversion after that, dropping the subject entirely, though there is a scene in the memoirs where a grateful village cheik tells him that he "talked like the Prophet".

Note that a simplified version of that story had appeared 43 years earlier in Walter Scott's The Life of Napoleon Buonaparte, Emperor of the French, published in 1827. Scott was given access to papers related to Saint Helena, so it is likely that his primary source was the same as the one used for the official version of 1870.

-> Part 2. What to make of this?