The Senate is stalling on bringing new Iran sanctions legislation to a vote following another diplomatic breakthrough, but pro-sanctions Senators say the House may not wait to pass the bill that the White House says could kill the talks and lead to war.

On Sunday, Iran and the P5+1 countries announced they had completed the implementation agreement for an interim deal regarding Iran’s nuclear program, starting the clock on a six-month period during which a potential final deal will be negotiated. Reports said today that there is also a 30-page informal side deal, known as a “non-paper,” that would spell out details of the interim agreement the parties don’t want made public.

The new progress in negotiations with Iran prompted Senate Majority Leader Harry Reid to say Tuesday the time was not right for a vote on the Menendez-Kirk Iran bill, which would set out Congressional parameters on what a final deal should look like and impose new sanctions if Iran does not complete the final deal or honor it.

"At this stage, I think we're where we should be,” Reid said, reversing his previous pledge that if the bill was bipartisan, he would bring it up for a vote on the senate floor. The bill currently has 59 co-sponsors, including 16 Democrats. Eleven Democratic Committee chairmen have also said they oppose moving forward to a vote at this time.

The White House has been battling those Democrats who support the legislation, accusing them of wanting war with Iran. House Minority Leader Steny Hoyer said Tuesday that the charge was “absolutely untrue” and an“irresponsible assertion” by White House staff.

Democratic National Committee Chairman Debbie Wasserman Shultz reportedly met with Hoyer to urge him not to bring up a non-binding resolution on the Iran deal with House Majority Leader Eric Cantor, a claim she denies.

Regardless, Sen. Lindsey Graham told The Daily Beast Tuesday he was working with Cantor to bring up the Menendez-Kirk bill in the House, where it will surely pass, as a way to pressure Reid to act.

“I am thinking of a House-first strategy,” said Graham. “We control the House, so I believe if we don’t get any movement from our Democratic colleagues over here, I would suggest the House take up this bill. I’m talking to Cantor.”

For Graham and other pro-legislation Senators, the most important part of the bill is the section defining what would be a good deal with Iran. They want the administration to negotiate a deal that would prevent Iran from continuing uranium enrichment, force Tehran to dismantle its nuclear reactors, and not allow the country to develop new advance centrifuges as part of any research and development program.

Congress goes on recess next week, but if Reid doesn’t act by the time Congress returns, Cantor should go ahead and call the vote, Graham said.

Senate Foreign Relations Committee Chairman Robert Menendez, who co-authored the legislation, told The Daily Beast in a short interview that he was not involved in the discussions with Cantor but he is not sure if Reid would respond to any House action.

“I don’t know that that would necessarily resolve the ultimate question but that’s something for the House to decide,” he said.

Menendez, like many Senators on Tuesday, said he was unsure if there was actually a secret side agreement to the interim nuclear deal, but he vowed to find out. Administration officials are set to brief Senators on the Hill on Thursday. Multiple Senators said staff briefings and calls by administration officials thus far on the deal have been cursory and unsatisfying.

“I don’t know for a fact but it seems to me that the Iranians saying there’s a side agreement suggest that there’s something,” he said. “If there’s no such thing, we need a denial, if there is we need to see the whole thing.”

At Tuesday’s State Department press briefing, State Department Spokeswoman Marie Harf said there was no such secret non-paper agreement.

“Let me be very clear, there’s no secret agreement here. The documentation associated with the implementation agreements tracks completely with what we’ve described, which are technical plans submitted to the IAEA,” she said.

Senate Armed Services Committee Chairman Carl Levin told The Daily Beast Tuesday the interim deal should be released in its entirety.

“We’re going to get a copy of it. I don’t know why it would be classified, it should be public,” he said.

But Levin is one of 11 Democratic Senate Committee Chairman opposed to moving forward with the Menendez-Kirk bill while negotiations are ongoing.

“It undermines the chances of getting a very strong agreement through diplomacy and negotiations with Iran,” he said. “It’s worth testing and worth trying and we ought to do that and not see that undermined by Senate or House actions.”

Sen. John McCain told The Daily Beast that the bill would only give the Obama administration more leverage in the negotiations, not destroy them as the administration claims.

“It’s hard to understand why they wouldn’t want to have an added incentive to get the deal that they are working on done,” he said.