President Barack Obama, escorted by Senate Sergeant at Arms Terry Gainer, waves as he arrives on Capitol Hill in Washington, March 12, 2013, to visit with Senate Democrats in the first of four meetings with lawmakers this week to discuss the budget. (J. Scott Applewhite/AP Photo)

Update at 4:55 p.m. ET -

White House Press Secretary Carney shot down a National Journal report quoting a senior administration official saying Obama's public outreach is a "joke" and a publicity stunt.

"I have no idea who said that, but I can tell you that opinion has never been voiced in my presence, in the President's presence, in the West Wing," Carney said. "It does not represent the President's view. It does not represent the White House's view, and it does not represent the administration's view."

President Obama met with Senate Democrats today for about 90 minutes behind closed doors during the Democrats weekly caucus lunch in a room just off the Senate floor in the U.S Capitol Tuesday.

The president will meet with Senate Republicans on Thursday. Senator McConnell said he welcomes the president's engagement - three separate trips to Capitol Hill this week and last week's dinner with a small group of a dozen Republicans. In Washington it has been dubbed Obama's "Charm offensive"

Democrats said after the meeting, the president took multiple questions, including on the hope for "some sort of grand bargain" to deal with the deficit, concerns about his willingness to accept changes to Medicare, and a proposal called chained CPI that would incrementally slow the growth of social security. At least one high level Democrats who expressed concerns about his drone program, according to Democrats present.

"We're looking forward to having him up on Thursday to meet with our group," said Senate Minority Leader McConnell said Tuesday afternoon. "I think the outreach that he's been having is a good thing. I'm all for it."

"The reports I got from the members who went down to dinner with him last week was excellent, that they had a good exchange," McConnell said, "I told the president on Friday I hope he'll invite all of our members down for these small dinners. It was a good opportunity to have a candid conversation."

One thing President Obama may hear a lot about when he faces Senate Republicans on Thursday will be his budget, which was announced today by the White House that it will be put forward on April 8 th, more than two months late.

Senator Mitch McConnell said the budget is "beyond tardy," and Sen. John Cornyn, R-Texas, intimated that President Obama is irrelevant in the budget debate.

"If there's a better way to render yourself irrelevant to the most - to the debate on the most important issue facing our country today, that may well be - that may well take the cake," Cornyn said, "because the president's budget will not even get here until after the House and the Senate have passed their own budget."

Original Post at 8:02 a.m. - President Obama kicks off three days of rare meetings on Capitol Hill today as he attempts to convince Congress to reach a deal to rein in the nation's deficit.

Obama will make the trip up Pennsylvania Avenue this afternoon for a meeting with the Senate Democratic caucus. Over the course of this week, the president will woo Republicans and Democrats in both the House and Senate as he tries to cut a deal.

While in the past, the president has focused on bringing his deficit reduction message directly to the American people, he is changing tactics and engaging rank-and-file lawmakers instead.

After Congress and the White House failed to reach a deal to avert $85 billion in across-the-board sequester cuts, and just months after the bitter battle over the "fiscal cliff," the president is hoping to capitalize on a brief lull before the next fiscal deadline.

"Now we have a period where, because of the choice to let the sequester take effect, we are not now in a countdown clock situation," White House Press Secretary Jay Carney explained last week. "There is an opportunity here to do what some members of Congress and leaders have said they would like to do, and we agree, and that is return to some sense of normalcy here, regular order, engage in a budget process and negotiation and debate that hopefully produces a bipartisan compromise."

Obama's outreach campaign, which began with dinner with Republican Senators last week and has been met with positive reviews, will be put to the test this week.

The president's meetings on the Hill come as Congress considers budget resolutions in both chambers.

House Budget Committee Chairman Paul Ryan, R-Wis., and Senate Budget Committee Chair Patty Murray, D-Wash., are both expected to release their budget blueprints for 2014 this week. The proposals, however, are expected to be far apart.

Carney said Monday that the president's discussions with lawmakers will not be budget negotiations. "I wouldn't expect that they're going to trade paper on numbers," he said. "[The president] looks forward to making clear what his policy positions are, to making clear his sincerity when it comes to his belief that we need to take action on our deficit."

The president is required by law to submit his budget to Congress no later than the first Monday in February, but Obama has yet to release his plan and the White House has not set a date for him to do so.

Also this week, Senate Democrats will introduce their bill to fund the government through the end of the year, after the House passed a version last week. Most federal agencies will run out of money March 27 if Congress doesn't act.

So could this charm offensive really work?

"I hope that this is sincere. We had a good, frank exchange. But the truth will be in the [coming] weeks as to whether or not it's a real sincere outreach to find common ground," Ryan, who had lunch privately with the president last week, said on "Fox News Sunday."

"If we're going to really get to an agreement, this is a good step," Sen. Ron Johnson, R-Wis., told George Stephanopoulos on ABC News' "This Week." "You have to start meeting with people. You have to start developing relationships. You've got to spend a fair amount of time figuring out what we agree on first."