Well, I think it's a given that it would either be a system where one of the two has all the power or a system where the power is split between the two. I think our founders would have favored the latter, but if the system was adapted later, like in the (quite plausible) Whig idea, the speaker of the house would probably have all the power.
I suppose the most likely scenario that doesn't start with the two heads being separate would involve congress passing a bill saying the president could only appoint secretaries that were first vetted by congress. The president would veto the bill, and congress would pass with a two thirds vote to override the veto. However, none of the people in congress would become the "Head of Government" unless one of the houses is more powerful than the other to begin with, in which case the head of that house becomes the head of government.
If it works this way, The president would still hold the veto power, but the cabinet would work for the head of congress. It would be different from a parliamentary system in that the people in the cabinet will not be members of congress.
Actually in quite a few parliamentary systems, members of the Cabinet aren't members of Parliament. They renounce their seats once appointed to an executive position.