Not quite on the Roman model of two consuls, but the Convention did debate whether to vest executive power an independent unitary executive or some kind of committee. My source is Madison’s Notes on the Convention.
The executive branch was first mentioned on May 29 as part of the Virginia Plan, but, at this point, there was no reference to the specific powers, composition, or selection of the executive. The next few days were largely subsumed with discussion of the legislature branch, but, on June 1, the composition of the executive branch was discussed.
Now, obviously, there was precedent for a unitary executive, but the framers did not need to go all the way back to Rome to find a model with multiple executives. At the time, the host of the convention, Pennsylvania, operated under a Supreme Executive Council, with a president, vice president, and representative from each county (11 at the time).
Turning to the debate on June 1 at the Convention. James Wilson initially proposed a strong unitary executive, which was seconded by Charles Pinckney.
Franklin, who had served as the president of Pennsylvania’s council expressed reservation about this and called for debate before voting for a unitary executive.
At that point, John Rutledge expressed support for a unitary executive, but did not want the executive to have power of war and peace, which would implicitly be reserved to the legislature as representative of the people’s will.
Roger Sherman took this a further and advocated a weaker executive with no fixed number. He proposed a plan wherein the legislature would appoint ad hoc executives to carry out the will of the legislature. In effect, the legislature could assign enforcement of the laws or specific laws to any number of individuals who were answerable to Congress.
At this point, Wilson apparently made an impassioned argument for an energetic unitary executive, while acknowledging that it should not be based exactly on the British model. He alluded to Rutledge’s point about the power of war and peace.
At that point, Edmund Randolph decried a unitary executive as a precursor of monarchy and suggested that a council of three would be as energetic as a unitary executive.
At this point, Madison suggested that the Convention address the powers of the executive, before determining its composition. The question about the number of executives was tabled.
On June 4, Wilson and Pinckney again suggested a unitary executive, with Wilson making another argument in favor. Sherman, who seems very wary of a strong unitary executive, then suggests a council of advice to constrain the executive.
Eldridge Gerry expressed that an executive by committee would be inconvenient in circumstances such as war.
At this point, the question was put to a vote and passed 7 to 3. The Convention then moved on to a host of other issues, mainly concerning the legislature. On July 17, the question of a unitary executive was revisited for final passage and approved 10-0. The Convention would address other aspects of the executive power, but at that point, it seems the Convention had settled on a unitary executive.
Advocates of a strong executive branch mentioned the consuls only as an example of how not to run a government. The complications caused by the dual magistracy of the consuls were brought up repeatedly during the Constitutional Convention. From Madison's notes on the proceedings, for example, we learn that:
"Mr. Wilson [James Wilson of Pennsylvania] spoke against executive plurality as unable to control legislation properly; he showed...how the division of executive power in the cases of the Roman consuls and the kings of Sparta resulted in factiousness."
As might be expected, the arch-Federalist Alexander Hamilton (who infamously proposed that the new government be an elective monarchy) was equally unimpressed with the consuls, and spoke admiringly during the Convention of the Roman custom of replacing the consuls with a sole dictator in times of crisis. Hamilton wrote to similar effect in Federalist no. 70, endorsing a strong executive:
"That unity is conducive to energy will not be disputed. Decision, activity, secrecy, and despatch will generally characterize the proceedings of one man in a much more eminent degree than the proceedings of any greater number; and in proportion as the number is increased, these qualities will be diminished. This unity may be destroyed....by vesting the power in two or more magistrates of equal dignity and authority....Of [this] the two Consuls of Rome may serve as an example."
Anti-Federalists, of course, were more inclined to see the benefits of a corporate executive. A(n admittedly brief) search of some of the more prominent Anti-Federalist tracts didn't turn up any explicit endorsements of a dual presidency, but Cato no. 4 observes:
"It is remarked by Montesquieu, in treating of republics, that in all magistracies, the greatness of the power must be compensated by the brevity of the duration; and that a longer time than a year, would be dangerous."
This is likely an allusion to the annual terms of the Roman consuls. (The eleventh book of Montesquieu's Spirit of the Laws, which many of the Founders had read, discusses the evolution of the Republic in detail.)
I talk a little more about the Founders' use of Roman history in this answer.