As an American, whenever we learn about the American and French revolutions (other contemporary revolutions like those in Haiti and South America are rarely mentioned, but I have personally looked into them) we occasionally have lip service paid to the influence of the English Civil War and many Enlightenment writers and philosophes like Rosseau or Hobbes as the source of republican fervor. Often, Europe is painted as being entirely controlled by monarchs, and while by the time of the American revolution much of Italy was ruled by some form of king, historically it had a variety of republics like that of Florence, Venice and Genoa.
Had the governments, philosophies and writers of these nations contributed to the republican thought that lead to the establishment of the American and French republics, directly or indirectly?
It's a great question. I'll do my best to answer the American side, as it's what I know best.
Three major books are the classic statements on this. Bernard Bailyn's Ideological Origins of the American Revolution first found that American revolutionary pamphlets were influenced by a strand of thought coming out of the English Civil War, which was itself informed by the writings of classical and Renaissance writers. Gordon's Wood's book The Creation of the American Republic showed how this "classical republican" ideology interacted with Enlightenment liberalism, and further developed the point. Finally, JGA Pocock's The Machiavellian Moment focuses on the influence of Florentine political thought throughout the revolutionary Atlantic, and particularly the role of James Harrington and Niccolo Machiavelli in disseminating that tradition. Pocock famously claimed that the American revolution, in adopting classical understanding of republicanism, virtue, and corruption (which he termed "civic humanism") was the last gasp of the Renaissance.
The answer to your question is that there was definitely a link between Italian Renaissance republicanism and American notions of republicanism. Whether this link was actually formative of republican thought, or if it was just one of many ideas swirling around, is something that scholars still debate.
Perhaps someone with more insight into the origins of the French Revolution can offer an answer on the French side. But my understanding is that the French Revolutionaries, compared to their American counterparts, were relatively quick to turn away from the influence of tradition on their thinking on political formation. The early French revolutionaries were more inclined to look at the republican traditions that preceded them (in the U.S. and Britain for example). I would not be surprised, though I admit I haven't read anything on this specific topic, if they were also interested in Italian republican thought.