President Barack Obama gestures as he speaks to reporters in the White House briefing room in Washington, Friday, March 1, 2013, following a meeting with congressional leaders regarding the automatic spending cuts. (Carolyn Kaster/AP)

After more than two years of failed negotiations with GOP leaders, President Obama is for the first time reaching out directly to rank-and-file Republicans who have expressed a willingness to strike a far-reaching budget deal that includes higher taxes.

In a flurry of meetings and phone calls over the past few days, Obama has courted more than half a dozen Republicans in the Senate, telling them that he is ready to overhaul expensive health and retirement programs if they agree to raise taxes to tame the national debt.

“He wants to do the big deal,” Sen. Lindsey O. Graham (R-S.C.) told reporters Tuesday after a 10-minute phone call from the president, their third conversation in the past two weeks.

Graham has repeatedly said he could support the White House’s goal of raising $600 billion in new revenue over the next decade in exchange for reforms to health and retirement programs. Now, he said, “what I see from the president is probably the most encouraging engagement on a big issue I’ve seen since the early years of his presidency.”

Sen. Susan Collins (R-Maine) said she, too, welcomed Obama’s call. “Even though it may be belated, the president does seem to be extending an olive branch and encouraging cooperation,” she said.

But there was more skepticism of Obama’s motives among other high-profile targets, including Sen. Saxby Chambliss (R-Ga.), a member of the Gang of Six, which labored for months on a bipartisan deficit-reduction deal without ever receiving the president’s attention.

“It’s hard to be too optimistic,” Chambliss said. While he considers the outreach “a good sign,” he said, “I don’t know that anything happened” to make Obama call now “other than his polling shows that a bipartisan, truly balanced approach is pretty popular out there in the country.”

It was not immediately clear whether the effort marked the beginning of a sustained White House campaign to circumvent Republican leaders on the budget or whether Obama is merely checking to see if there’s any appetite for undoing the sharp automatic spending cuts known as the sequester.

The cuts hit Friday and will slice $85 billion out of agency budgets this year without an agreement to cancel or replace them.

On Tuesday, White House press secretary Jay Carney said that Obama is interested in “assembling a caucus of common sense and working with them to bring about a resolution to this challenge.”

Although the conversations have focused largely on the budget, Carney said, the president is talking with lawmakers about “a variety of issues, not just our fiscal challenges.”

For weeks, Obama has pleaded with GOP leaders to revisit a plan he put on the table in December that would raise roughly $1.8 trillion over the next decade through a combination of tax increases, Medicare reforms and reductions in the annual cost-of-living adjustment for Social Security benefits.

Alan Kreuger, chairman of the White House Council of Economic Advisers, pressed the case for Obama’s “balanced” approach again Tuesday in a speech to the National Association for Business Economics.

But House Speaker John A. Boehner (R-Ohio) and Senate Minority Leader Mitch McConnell (R-Ky.) have rebuffed the president’s entreaties, saying they will not agree to replace spending cuts with tax increases. During a meeting with the four top congressional leaders Friday at the White House, Obama broached the issue of entitlement reform but, according to an aide familiar with the meeting, found no takers.

So on Saturday the calls to rank-and-file GOP senators began, starting with Sens. Bob Corker (Tenn.), Rob Portman (Ohio) and Chambliss. On Monday, Obama spoke to Collins and to Sen. Tom Coburn (Okla.), who said the two discussed the negotiations over enhanced background checks for gun purchases as well as budget issues.

Obama had previously spoken to Sen. John McCain (R-Ariz.) in a meeting at the White House with Graham that focused primarily on immigration.

Meanwhile Tuesday, the White House said that Obama is “deeply concerned” about the impact of a House Republican funding measure that would avoid a government shutdown later this month but lock in the sequester for the rest of the fiscal year. But the statement stopped short of threatening a veto, and congressional leaders expressed optimism that a showdown will be averted.

“There seems to be no interest on either side in having a kind of confrontational government- shutdown scenario,” McConnell said.

Phil Rucker contributed to this report.