From what I remember (I did a masters in history about a decade ago), Eran Shalev's _Rome Reborn on Western Shores_ was the main book on the subject (edit: the subject being Rome / the classics and early American political culture generally, not specifically the US Constitution).

That said, there was some debate over the influence of Rome on early American political debates, with others citing either "classical republicanism" or the pre-English Civil War debates on the English constitution as more important influences (for example, JGA Pocock or Gordon Wood).