One day after President Obama submitted his budget request for fiscal 2012 to Capitol Hill, congressional Republicans assailed the document as too weak on spending. But they also signaled an openness to working with Democrats to solve the nation’s financial problems.

The mixed message reflects uncertainty among resurgent Republicans about how best to address the concerns of voters — and satisfy the demands of restless freshman members of their own party.

Some Republicans, particularly those in the House, want to force an immediate showdown with Democrats: GOP leaders have included sharp cuts to federal agencies in a must-pass spending measure that would keep the government open through September. Other Republicans, including many longtime senators, want to seize the moment to join Democrats in overhauling politically sensitive programs such as Social Security and Medicare, the biggest drivers of future spending.

The confusion was evident Tuesday in congressional budget hearings, where GOP lawmakers grilling White House budget director Jacob J. Lew veered between criticizing Obama as not addressing the looming crisis in entitlement programs and asking what they could do to help.

“People on this panel here, at least on this side of the aisle, invite the dialogue, encourage it, with the White House on this,” said Rep. Todd C. Young (R-Ind.), a Marine vet­eran who sits on the House Budget Committee. “What might I do, as a freshman member of Congress, to create the political space where the president can step up and take a leadership role in these matters?”

At the White House, Obama played down his much-criticized $3.7 trillion spending plan in his first news conference of the year, calling it the initial step in what promises to be a lengthy negotiation over the government’s fiscal future. He again invited Republicans to join him in seeking bipartisan solutions for handling the skyrocketing costs of health care and an aging population.

“The first step in this budget is to make sure that we’re stabilizing the current situation,” the president said of his blueprint, which would reduce record budget deficits over the next decade and stabilize government borrowing. “The second step is going to be to make sure that we’re taking on some of these long-term drivers.”

For now, at least, those steps are tangled in a single battle that will affect spending levels this year, next year and beyond. Untangling the process, Obama said, will require both parties to engage in the kind of trust-building that produced legislative successes in December, including an extension of dozens of expiring tax breaks, repeal of the “don’t ask, don’t tell” ban on gay men and lesbians serving openly in the military, and the New START nuclear pact.

“This is not a matter of ‘You go first’ or ‘I go first,’ ” Obama said. “This is a matter of everybody having a serious conversation about where we want to go and then ultimately getting in that boat at the same time so it doesn’t tip over.”

In the Senate, where the December deals were forged, Republicans were receptive to that message. But they argued that Obama missed a chance in his budget to send a signal that costly entitlement programs are on the table.

“Entitlement reform will not be done except on a bipartisan basis with presidential leadership,” Senate Minority Leader Mitch Mc­Con­nell (R-Ky.) told reporters. “I’ve been inviting the president to have that conversation since he took office two years ago,” he added. “It doesn’t have to be in public. We all understand there are some limitations to negotiating significant agreements in public.”

Sen. Lindsey O. Graham (R-S.C.), a longtime advocate of bipartisan entitlement reform, echoed that view in a Budget Committee hearing.

“Tell me where I need to go and who I need to meet with to find a way to save Social Security,” he demanded of Lew. “This is the year. There are a lot of Republicans who understand entitlements have to be put on the table.”

The Senate is fertile ground for ambitious talks. A group of senators from both parties is working to advance a budget strategy offered last year by Obama’s fiscal commission, which won the support of key Senate liberals and conservatives. That proposal — which calls for politically perilous moves such as raising the retirement age, charging wealthy seniors more for Medicare and eliminating cherished but expensive tax breaks — would save $4 trillion over the next decade, four times the savings in Obama’s 10-year budget projections.

The prognosis is less clear in the more conservative House, where Speaker John A. Boehner (R-Ohio) is struggling to corral an independent-minded caucus packed with 87 newcomers. For now, House leaders are focused on the short term and on fulfilling their campaign pledge to slash government spending this year.

On Tuesday, the House began debating a temporary spending bill that would fund the government through Sept. 30 but provide $61 billion less than agencies had been expecting. House leaders plan to hold a final vote on the legislation by week’s end, when they are scheduled to leave for a week-long recess. More than 400 amendments to the measure have been filed, and GOP leaders said the chamber will vote on as many as possible.

The current continuing resolution, approved in December, will expire March 4, leaving Congress little time to resolve vast partisan differences. Democrats have accused Republicans of taking a “meat cleaver” to the government, seeking “draconian” cuts that could damage vital services, eliminate hundreds of thousands of federal jobs and destabilize the economic recovery.

Meanwhile, Republican members of the House Budget Committee hammered Lew for Obama’s inability to deliver the immediate cuts they say their constituents are demanding. Although House Republicans want to cut $61 billion from domestic programs this year, Obama’s budget would trim just $6 billion from such programs — and would do so next year.

They also complained that Obama has not eliminated budget deficits, setting an exceptionally high standard for the spending plan that House Budget Chairman Paul Ryan (R-Wis.) is due to assemble this spring.

After Obama’s news conference, House leaders vowed in a statement that Ryan’s plan will offer a vision for tackling entitlement programs. “Our budget will lead where the president has failed, and it will include real entitlement reforms so that we can have a conversation with the American people about the challenges we face and the need to chart a new path to prosperity.”

Staff writers Brady Dennis, Paul Kane and Felicia Sonmez contributed to this report.