President Barack Obama Tuesday launched an unusual mission to woo lawmakers on Capitol Hill, but budget clashes with Republicans underscored the difficulty of striking a grand deficit bargain.

Obama trekked up Pennsylvania Avenue for the first of three visits, flashing a broad smile as he went into lunch with Democratic senators, to push his vision for a deficit cutting deal based on more tax revenue and spending cuts.

In the next two days, the president will enter the lion's den in separate talks with Republicans from both chambers and will also meet minority lawmakers from his own Democratic Party in the House of Representatives.

Obama has often appeared to disdain the back-slapping and arm-twisting that greases power between a president and Congress, but has made a new effort in recent days, and dined with a dozen Republican senators last week.

Senate Republican minority leader Mitch McConnell, who has used his mastery of procedure to gum up Obama's legislative program, complimented the president for his effort, though offered no real signs of a breakthrough.

"With regard to what a lot of you have described as the president's charm offensive, we welcome it," McConnell told reporters.

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

Democrats said the president mostly spoke about deficit cutting and budget issues and also expressed optimism that a bipartisan deal to fix America's broken immigration system could be reached.

"He thinks it's very important that we solve these problems together and he says that working together with Republicans in terms of getting a grand bargain, or a major dent in this issue, is critically important," said Democratic Senator Carl Levin.

But "compromise is essential and he hasn't seen enough of it from (Republicans) yet," Levin said.

Democratic senator Tom Harkin said several senators brought up Republican plans to cut spending on social programs, adding: "We're cautioning about that, be careful about this grand bargain."

While atmospherics were improved, there were signs Tuesday, as Republicans and Democrats unveiled rival budgets, that staunch divisions over basic political philosophy could derail the latest push for compromise.

White House spokesman Jay Carney called a new budget released by Republican House budget committee chairman Paul Ryan the opposite of the balanced blend of extra tax revenue and spending cuts the president wants to see enacted.

"While the House Republican budget aims to reduce the deficit, the math just doesn't add up," Carney said.

"By choosing to give the wealthiest Americans a new tax cut, this budget as written will either fail to achieve any meaningful deficit reduction, raise taxes on middle class families by more than $2,000 -- or both."

He also said that the Republican approach would turn Medicare, the popular health insurance program for the elderly, into a voucher system -- an approach Democrats say would lead to benefits falling way behind healthcare costs.

"We've tried this top-down approach before. The president still believes it is the wrong course for America," Carney said.

Ryan, the Republican party's 2012 vice presidential candidate, said his budget would alleviate the "crushing burden" of debt that is threatening America's future and would cut $4.6 trillion in spending.

The plan contains no new tax revenue and demands massive spending cuts, as well as major changes to cherished social programs like Medicare and Medicaid, in a bid to balance the budget within a decade.

Ryan on Tuesday described his plan as an "invitation to the president of the United States and to Senate Democrats to come together to fix these problems."

"Show us how to balance the budget," he said.

Democrats in Congress are unveiling their own budget plans, which include nearly $1 trillion in new revenues and $1 trillion in cuts.

The White House, under fire from Republicans over delays in its own budget blueprint, predicted Obama's own plan would emerge after April 8.