President Barack Obama on Tuesday called three of the four Republican senators working with four Senate Democrats to draft comprehensive immigration legislation that they hope to unveil next month.

“This afternoon, the President placed calls to [the senators] to discuss their shared commitment to bipartisan, commonsense immigration reform and to commend the Senators for the bipartisan progress that continues to be made by the Gang of 8 on this important issue,” the White House said in a release.

Obama spoke with John McCain of Arizona, Lindsey Graham of South Carolina and Marco Rubio of Florida. The fourth GOP member of the group, Jeff Flake of Arizona, was traveling, but the White House said Obama “looks forward to discussing the issue with him in the near future.”

Obama met with Democratic members last week.

Rubio spokesman Alex Conant tweeted that the senator “appreciated [the president’s] call to discuss immigration tonight. Rubio said he feels good about ongoing negotiations in Senate.”

Earlier in the day, White House Press Secretary Jay Carney disputed charges from Conant that the White House has not reached out to the GOP.

“President Obama and the White House staff are not working with Republicans on immigration reform,” Conant said in an email to reporters, disputing the White House’s account. “Senator Rubio’s office has never discussed immigration policy with anyone in the White House. To be clear: That’s fine with us — we actually think Congress should write the policy; not the White House.”

Carney, asked about the comment, insisted that the White House has been in meetings with the staffs of members of the “gang of eight,” of which Rubio is a member.

In the readout of the White House’s afternoon calls to GOP senators, the White House carefully noted that the calls “build on conversations that have taken place at the staff level.”

The White House said the president reiterated to Republicans “that he remains supportive of the effort under way in Congress, and that he hopes that they can produce a bill as soon as possible that reflects shared core principles on reform.”

“The President has made clear that he believes commonsense reform needs to include strengthening border security, creating an earned path to citizenship, holding employers accountable, and streamlining legal immigration,” the White House continued. “As the President made clear when he met with Democratic Senators involved in the process last week, that while he is pleased with the progress and supportive of the effort to date, he is prepared to submit his own legislation if Congress fails to act. He thanked the Senators for their leadership, and made clear that he and his staff look forward to continuing to work together with their teams to achieve needed reform.”

Indeed, the White House has already begun crafting its own bill. Carney said a leak of details of the White House’s own immigration bill this weekend was not intentional. He said the administration prefers and hopes a bipartisan deal emerges from the Senate.

Carney had earlier struggled to explain why Obama hasn’t picked up the phone or met with GOP leaders this year to try to resolve the dispute over automatic spending cuts, known as the sequester, amid broader questions about the administration’s congressional relations.

Congressional Republicans have been peeved by the White House’s campaigning across the country instead of sitting down to negotiate.

Carney, asked repeatedly Tuesday why the president has not called Senate Minority Leader Mitch McConnell, R-Ky., did not give a direct answer, instead throwing up a vague response that the administration is interacting with Congress on a range of issues.

As for meeting with GOP leaders on the sequester, Carney said, “The President has made clear his door is open. What we’ve heard from the Republicans thus far is a categorical refusal, at least at the leadership level, to accept the basic principle that balance is necessary.”

Pressed further, Carney added, “The problem here isn’t a lack of meetings around the table in the Roosevelt Room or the Cabinet Room.” He noted that in the past Speaker John A. Boehner of Ohio has been the one who pulled out of talks on budget issues.

Shortly thereafter, via Twitter, spokesmen for McConnell and Boehner tweeted to Buzzfeed’s Zeke Miller that the president hadn’t spoken to their bosses this year or since the inauguration, respectively.

Carney later in the briefing would neither confirm nor deny meetings between leaders and the president this year.

“I’m not suggesting anything about what’s happening now. I’m just saying that we don’t always read out every conversation or meeting we have with members of Congress,” he said. “And in taking that approach, we’re mindful of the requests of others as well as our own interests.”