The Founding Fathers of the Untied States certainly were interested in Rome, but they almost universally held the belief that they'd have to learn from Rome's weaknesses. The Roman Republic collapsed into civil war and empire, after all. The issue is that they drew widely divergent conclusions from this history. Some believed that there had been too great an opportunity for one man to cease overall power-- and so they favoured a weak central government, short terms of office (like a year, or even six months), and often felt positive about collegial executives (like a council of five instead of a president). Others believed that only a strong leader could prevent chaos (they essentially believed Caesar had been the right solution for Rome), so they wanted a strong central government, life-time terms in office, and a single chief executive with broad powers.
While there were certainly moderates in between these extremes, very few people believed directly emulating the Roman Republic was a good plan. As has been noted: if there were going to be multiple executives ruling together, there would be three of them, so that you could prevent a tied vote.
The only way I see a system of two presidents coming about is if the Constiution (or an analogy to it) is rejected by a number of states, and they choose to keep operating under revised(!) Articles of Confederation that provide for a Chief Executive. Assuming that the "Confederal states" and the "Federal states" would keep foreign policy and the military a shared concern, the two presidents of these two unions would share responsibility for those matters. The USA would simply be a "union of two unions".
