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Update at 5:36 p.m. ET -

President Obama made his first direct overture to Republicans on immigration tonight, placing calls to the three key GOP players on the issue in the Senate: John McCain, Lindsey Graham and Marco Rubio.

The calls come after Rubio called a White House immigration plan that leaked over the weekend "half-baked and seriously flawed".

Rubio says he appreciated the call (even though he may have been roused from bed to take it; he's in Israel):

"Senator Rubio appreciated receiving President Obama's phone call to discuss immigration reform late tonight in Jerusalem," according to a statement from Rubio's office. "The Senator told the President that he feels good about the ongoing negotiations in the Senate, and is hopeful the final product is something that can pass the Senate with strong bipartisan support."

Original Post at 3:23 p.m. ET -

So, is the White House talking to Republicans on immigration or not?

The leading Republican on the issue is Senator Marco Rubio, R-Fla., and his office told reporters today that they've had no consultation with the White House on the issue of immigration.

"President Obama and the White House staff are not working with Republicans on immigration reform," Rubio spokesman Alex Conant said in an email to reporters. "Senator Rubio's office has never discussed immigration policy with anyone in the White House."

White House Press Secretary Jay Carney disputed that. "We have been in contact with everybody involved in this effort on Capitol Hill," he said.

In fact, the White House claims the president's aides have had no fewer than five meetings with the staffs of the bi-partisan group of Senators working on an immigration bill - including Rubio's staff.

Back to you, Rubio spokesman Alex Conant.

"We have never discussed immigration policy with anyone from the White House," says Conant. "The administration has sent some agency officials to brief staff at the bipartisan group meetings, but they've never asked for our input. We've never received a call or email from [Director of the White House Policy Council] Cecilia Munoz or anyone else at the White House asking for our input as they draft their bill."

So, maybe it has been a one-way conversation?

At any rate, it can't be a good sign for immigration reform if the two sides cannot even agree on who has talked to who about what.