I believe that there is no good evidence that any serious proposition was ever made to make Washington a monarch. Given that he appears to have been shooting blanks, he would have had no dynastic successor, so the effort would have foundered at his death. In fact, he stepped down at the end of his second term in office precisely because he did not wish to be "President for life" and to annoint his Vice President as successor for life--we know this because it is to be found in more than one place in his correspondence.
There was an organization formed after the revolution known as the Society of the Cincinnati. These were mostly former officers of the Continental line, although members of the militia were well-represented. Mostly, these were disaffected people who felt that the nation owed them more.
The society was named after Lucius Quinctius Cincinnatus, the scion of an old and venerable family of Rome, of the order of
Patres. His nephew had been impeached for treasonous correspondence with the Veiians (then the principle Roman enemy) and Cincinnatus pledged his very valuable property as bond for the appearance of his nephew in court, reserving only a small farm on the Janiculum (a hill across the river Tiber to the north, it was then uninhabited, and the crown was the site of the Campus Martius, where the city legion assembled and drilled). His nephew
had been in treasonous correspondence, and he absconded--which made all of the property of Cincinnatus forfeit. So, he retired to the Janiculum to become a simple farmer and continue to support his wife and himself (his children were grown and out on their own). As a member of the Quinctii, he would have been honored in any event, but this rectitude of behavior was such that he was even more revered, and it is said in the history of Titus Livius that he served five times as Dictator (an emergency office) and always behaved with probity, surrendering the
fasces, the symbol of office, as soon as the emergency had passed. This is crucial, because only the Dictator could carry a fasces with an axe bundled in it (his aides, the lictors actually carried the fasces), meaning he had the power of life and death, the ultimate power. The image below is of a statue in Cincinnati, Ohio, showing Cincinnatus surrendering the fasces preperatory to returning to the plow.
The Society of the Cincinnati offered the presidency of their organization to Washington. All of his life, Washington had railled against "faction" (what we would know as political parties and smoke-filled rooms), and he demonstrated once again his probity, refusing the office and publicly condemning the organization. The Society of the Cincinnati, at their web site (yes, they still exist, and they have a web site) claim that he was their first president. They carefully word it, that he was the first person elected president, and they don't say that he turned the office down.
Washington's refusal to join and lead the organization and his public condemnation of the organization and its goals lead them to change the name of their new settlement on the Ohio River from Fort Washington to Cincinnati. I think that not only is there no evidence that Washington ever sought monarchical power, but that he would not have accepted it had it been offered to him.