According to today's Times, they are already thinking about this, with Jason Furman consulting with Congressional "Blue Dogs" about budget rules, "including a potential law requiring balanced budgets to formalize the pay-as-you-go approach favored by the coalition."
I'm increasingly convinced that balanced budgets and annual pay-as-you-go aren't really the right way to go about this. The focus should instead be on a combination of short-term (five or ten year) plus long-term (infinite horizon) balance, or at least constraints on making things worse, given the distinct political pathologies that can lead to violation of either. There are also tricky design questions involving the choice between super-majority rules and automatic changes (such as sequesters) if targets are missed.
More on this as things develop if it gets anywhere. But the basic underlying task of designing constructive budget rules is tricky to say the least, and more art than science given the political economy aspects.