The votes would be unlikely to reach the floor in the Democrat-controlled Senate. Still, they would mark a break from the House’s recent Obamacare strategy, in which votes on the full repeal of the bill have taken precedence over even politically useful piecemeal changes. The preference among some members for an all-or-nothing strategy was laid bare in April, when Cantor failed to secure votes for a bill to transfer Obamacare funds to extend a special insurance program for people with preexisting health conditions. He had listed the bill, the Helping Sick Americans Now Act, as a top legislative priority. But revolt by a group of House conservatives and outside groups sank the effort. It was pulled from the floor and has not reappeared. A third full repeal vote, however, did pass the chamber. The individual mandate is seen as such a central provision of the law that it may be an easier sell to the caucus. “If you’re trying to kill the patient, full repeal is vaporization,” said a GOP House leadership aide. “But this is removing the liver.”

Dan Holler, the communications director at Heritage Action, one of the groups that strongly opposed the Cantor bill, is more lukewarm on the current Boehner strategy. “If you’re serious about doing more than scoring political points, if you’re serious about taking the entire law off the books, you need to defund the entire law,” he said, emphasizing that his group wants a full repeal. Still, he said, it has not yet decided to actively oppose the new strategy. The conservative Club for Growth, another influential opponent of the Cantor bill, declined to comment.

When asked if Democrats would be put in an awkward position by such votes, Rep. Bill Pascrell, D-N.J., said, “Absolutely not. They’re two different species.”