That will by no means be easy. Even if the two-thirds threshold were reached, a convention would probably face a court battle over whether the legislatures’ calls for a convention were sufficiently similar. And as with any amendment that Congress proposes, state-written amendments would need approval by three-quarters of the states — either by their legislatures or by state conventions — to take effect.

But as Republicans have surged to control of state legislatures and moved sharply rightward during the Obama years, what was once a pet project of the party’s fringe has become a proposal with a plausible chance of success. Some of the former Republican presidential candidates, including comparative moderates like John Kasich and Jeb Bush, have endorsed a state amendment convention.

So far, 28 states have adopted resolutions calling for a convention on a balanced-budget amendment, including 10 in the past three years, and two, Oklahoma and West Virginia, this spring. That is just six states short of the 34 needed to invoke the Article 5 clause.

“I think the prospect is very good in 2017,” said Gary Banz, a Republican who is the majority whip in the Oklahoma House of Representatives. “You can look at any number of states that are not on board yet, and they’re controlled by very conservative elements.’’

Including nominally nonpartisan Nebraska, Republicans now control 31 state assemblies — more than double the number in 2010. Of the 11 states advocacy groups have targeted for pro-convention lobbying next year, Republicans control both houses of the Legislature in seven.