I think America was often hugely inspiring to some contemporary onlookers. The idea of natural rights was very alluring to French Revolutionaries across the Atlantic. I'm not sure how much impact specifically the constitution had, but the general intellectual legacy of the American War of Independence was definitely important. The Declaration of the Rights of Man published by the French National Constituent Assembly in 1789 was directly worked on by Lafayette and Thomas Jefferson in joint communication with one another for example, while Benjamin Franklin was still alive and writing letters to friends in France. Robespierre was inspired by the American example to argue for freedom of the press. As Scurr notes however,

It [the Declaration of the Rights of Man] differed considerably from the American - it was more condensed, more abstract, and more suited to France's peculiar circumstances [i.e. Ancien Regime feudalism and strong hierarchical distinction].

So the international reaction was definitely there, but that didn't lead to an exact copy of what the Americans were doing, at least in terms of France.

Some stuff: The Coming of the French Revolution, Georges Lefebvre. Fatal Purity by Ruth Scurr