The major sticking point of a comprehensive agreement will be taxes. Obama and Democrats want to eliminate tax breaks and loopholes worth about $600 billion over 10 years as part of a broader $1.2 trillion deficit-reduction package that would include entitlement reforms.
Some Republicans have indicated support for ending such tax breaks as part of a broad deal. However, the fiscal-cliff agreement in January returned tax rates on top income earners to higher levels of the 1990s, and GOP leaders now oppose any further steps to raise rates or tax revenue.
A sticking point in a possible compromise on taxes would be whether increased revenue realized through reforms, such as eliminating existing loopholes, go towards holding down rates or reducing the deficit.
Meanwhile, Republicans say Obama and Democrats must deliver on significant entitlement reforms.
Sen. Rob Portman, R-Ohio, argued Sunday on CBS that Obama must use his profile to convince fellow Democrats and the American people that changes to Medicare and other entitlements are necessary.
"What the president needs to do is reach out not just to Republicans but to Democrats and to ensure that he gives them the political cover to do frankly what most of them know needs to be done," Portman said.
One of the GOP senators who dined with Obama last week said the president showed that he understood the scope of the problem, with Medicare paying out $3 in benefits for every $1 put in.
"I think he gets it," Sen. Tom Coburn of Oklahoma told NBC on Sunday. "And I think he`s genuinely reaching out. But you`ve got a lot of scabs and sores on people that it`s going to take a while for that to heal."
Another entitlement reform Obama has proposed would tighten the adjustment for inflation of benefits such as Social Security, meaning annual increases for future recipients would grow at a slower pace.
Opponents of the reform, known as "chained CPI" in reference to the consumer price index it involves, argue it hurts vulnerable senior citizens and others who most need their benefits.
If achieved, a grand bargain would give Obama a major political victory and a boost in cementing his desired presidential legacy after the controversial health care and Wall Street reforms of his first term.
Republicans also would get credit from moderates and independents for a willingness to compromise, but conservatives could punish them with primary challenges in 2014 and beyond.
After the repeated failure of previous efforts for a broad deficit reduction deal, attention has focused on a more limited agreement that would include some elements under discussion.
For example, a smaller agreement might end some tax breaks and loopholes while cutting Medicare costs paid providers, not beneficiaries, to achieve $500 billion or so in deficit reduction over 10 years.
Such an outcome, coupled with previous spending cuts and the January fiscal-cliff deal, would fail to reach the total $4 trillion in deficit reduction over the next decade that economists and political leaders have targeted as the minimum amount needed.
It also would allow both parties to simultaneously claim credit for making some progress after the past years of dysfunction while continuing to blame the other for preventing more.
The status quo outcome would mean continued brinksmanship over each pending fiscal deadline, as well as further economic uncertainty that already has lowered the U.S. credit rating and slowed growth.
For now, political leaders sound optimistic that resolve to move forward exists on both sides.
"I think there are things that we can do that don't offend either party's philosophy, that doesn't require someone to surrender their principles to make a good down payment on getting this debt and deficit under control," Ryan said Sunday.
His budget proposal will include provisions certain to be rejected by Democrats, such as eliminating 2010 health care reforms that comprise Obama's signature legislative achievement, as well as shifting Medicaid to a grant program for states and changing Medicare from a direct benefit to a direct subsidy program.
"We think that's the best way to make these programs work better, but are there things you can do short of that, that gets you closer to balancing the budget, that delays the debt crisis from hitting this country?" Ryan asked. "Yes, I think there are."
Obama's Medicare plan focuses mainly on reducing payments to drug companies and hospitals, though he would also raise revenue by asking wealthy seniors and new beneficiaries to pay more.
Democratic Sen. Tim Kaine of Virginia, a former governor who joined the chamber this year, told NBC that returning to a normal budgeting process creates a framework for debate and compromise.