Rep. Eric CantorEric CantorIf we want to make immigration great again, let's make it bipartisan Top Lobbyists 2017: Hired Guns GOP Rep. Jeb Hensarling to retire after end of current term MORE (R-Va.), the second-ranking Republican in the House, tied the use of budget reconciliation rules on the healthcare bill to Democrats' electoral fortunes this fall.

"It is my belief that the Democrats will lose their majority in November if they ram this thing through without any bipartisan support," Cantor said during an appearance on Fox News.

The GOP has been strutting with some confidence about their chances of taking back the House this fall, though Cantor's words may be the most direct link to date by a Republican leader between procedure on the healthcare bill and this fall's midterm elections.

At issue is the likely use of budget reconciliation rules to finish up work on a healthcare package that has consumed Congress for months. The plans under consideration would see the House pass the healthcare bill approved by the Senate in December, followed by a separate measure passed by the House to make changes to that first bill.

The package of fixes would be passed under reconciliation rules, meaning Democrats could pass the measure with a simple majority vote, instead of the 60 votes needed to head off a filibuster.

Republicans have railed against the measure as a partisan maneuver, and Cantor said Democrats would "unquestionably" lose as a result of the tactic.