With White House pushback against an Iran sanctions bill keeping the Senate at bay, House GOP leaders are considering taking up the Senate version of the bill this month to pressure the chamber to act, according to a source familiar with GOP leadership and committee thinking.

The discussion comes only a day after President Barack Obama announced the first step in a multilateral agreement mandating that beginning Jan. 20, Iran would begin dismantling its higher-level enriched uranium and enrichment infrastructure.

The latest breakthrough, announced in a White House statement on Sunday, is giving senators further pause about moving their bill, and support has topped out at 59 co-sponsors, one shy of the threshold needed to pass the bill without the threat of a filibuster.

The House overwhelmingly passed their own sanctions bill last year, but that was before diplomatic efforts yielded an interim agreement with Tehran to limit its nuclear ambitions. Now to spur the Senate along, House GOP leaders could schedule the Senate language for a vote later this month.

“Up until this week we wanted to hold firm,” the GOP source said. “But we think if we get that done in the House, that adds to the momentum and it can get to the president’s desk.”

Obama has threatened to veto any legislation implementing new sanctions, claiming that it could jeopardize the diplomatic process.

Proponents of the legislation, however, argue that the threat of sanctions would give Obama and his diplomats an important bulwark in case Iran goes back on the agreement.

“I’m concerned that this agreement takes us down that path where sanctions pressure is relieved, but Iran maintains its ability to produce a nuclear weapon,” House Foreign Affairs Chairman Ed Royce, R-Calif., said in a Monday statement. “Given these stakes, it’s regrettable that the President does not want to work with Congress to bolster his negotiating hand with additional sanctions, which would go into effect should Iran fail to meet its commitments.”

It remains an open question whether the House and Senate have the two-thirds majority in each chamber needed to override a presidential veto, if it comes to that.

Either way, action in either chamber would pit powerful Israel-backing groups pushing for sanctions against the president in a battle for Democratic support, a rift that could split Democratic political interests early in a congressional election year.

The six-month interim agreement struck in November would ease sanctions on Iran from the U.S., Russia, China, England, France and Germany if Tehran begins limiting its nuclear capabilities.

The Senate bill, co-sponsored by Foreign Relations Chairman Robert Menendez, a New Jersey Democrat, and Mark S. Kirk, an Illinois Republican, would give Obama more than a year to continue talks before imposing new sanctions on Iran’s oil exports and other parts of its economy should the country go back on the deal.