President Obama has been “getting absolutely no sleep” amid tireless work on the debt-ceiling crisis, according to senior aide Valerie Jarrett. No, countered press secretary Jay Carney, the president seemed “well rested and very focused.”

The apparent contradiction illustrated the White House choice as the budget battle with Congress nears its climax: whether to present Obama as a central player asserting his power to craft a deal, or as an above-the-fray observer waiting for a polarized Congress to do its job.

How the president navigates that balance could determine whether the outcome of the crisis proves a political plus for him or a drag that further endangers his reelection prospects.

Obama reentered the public debate Friday after three days out of view, using remarks in the Diplomatic Reception Room to call again for bipartisan compromise and asking Americans to pressure lawmakers through phone calls, e-mails and online activism.

“Tweet,” the president said. “Keep the pressure on Washington.”

Obama dispatched White House aides, including Office of Management and Budget Director Jacob J. Lew and Treasury Secretary Timothy F. Geithner, to meet with lawmakers on Capitol Hill.

And though Obama said Friday that the “time for putting party first is over,” he enlisted help from his 2012 reelection campaign — which sent to its 9.4 million Twitter followers the names of GOP lawmakers to be targeted for pressure and began a log tabulating calls by supporters to Republican offices.

White House officials insisted Friday that Obama was deeply involved all week, despite his low profile. He had canceled fundraising events and cleared his schedule, and as Jarrett told Reuters, has been “working tirelessly, meeting with his economic team, doing a lot of outreach, exploring all kinds of possibilities for compromise.”

The Friday effort followed three days in which Republicans regularly accused Obama of shirking his leadership duties and some Democrats groused privately that he was not putting enough of his own political capital on the line.

Obama’s public presence in the debate has wavered from week to week.

Earlier this month, the president appeared eager to be the chief broker — hosting lawmakers for repeated West Wing meetings. In a flurry of televised appearances, Obama veered between demanding bipartisan cooperation and likening Congress members to procrastinating schoolchildren.

Since Obama’s negotiations with House Speaker John A. Boehner had collapsed July 22, the president had begun to look sidelined in the negotiations as Boehner (R-Ohio) and Senate Majority Leader Harry M. Reid (D-Nev.) hashed out their own plans.

But when Boehner delayed a vote late Thursday on a plan that would slash $1 trillion through spending cuts and temporarily lift the debt ceiling — stymied by tea party conservatives demanding more cuts and a balanced-budget amendment — the White House saw an opening to reenter the debate.

Democrats also began to view the House GOP’s struggles as a sign that the tea party movement, a galvanizing force for Republicans in the 2010 elections, may prove more complicated for the GOP next year. Conservatives remain energized, particularly in their quest to defeat Obama. But Democrats now hope to accuse the GOP’s tea party wing, along with the presidential candidates appealing to its members, of pushing the country to the brink of financial crisis.

Reid on Friday railed against conservatives, calling them “bizarre.”

David Axelrod, Obama’s campaign strategist, said in an interview that the debt-ceiling battle was shaping up as a “definitional fight” in which voters would perceive the president as pursuing a reasonable middle ground and Republicans “pandering to the extremes.” He noted that GOP presidential front-runner Mitt Romney endorsed the House GOP’s “cut, cap and balance” plan.

“In the values and the goals that have been staked out in this debate, you’ve got a really stark contrast,” Axelrod said. “That contrast is going to help define both the president and the opposition moving forward.”

Andrea Saul, a Romney spokeswoman, charged that the debt ­crisis “started with President Obama’s failure to lead, and now we’re embarrassed in the eyes of the world.”

In the short run, at least, it appears the debt-ceiling fight has not been helpful for Obama’s political prospects.

New polling numbers suggest that voters have been unimpressed with Obama’s performance — with his once-sizable reelection advantage evaporating in a matter of weeks.

A survey published this week by the Pew Research Center for the People and the Press finds that 41 percent of voters want to see Obama reelected next year, compared with 40 percent who favor a Republican. In May, Obama led by 11 points, 48 to 37. The explanation: The number of independents wanting an Obama victory fell from 42 percent in May to 31 percent now.

One Democrat facing reelection next year, Sen. Joe Manchin (W.Va.), took some swipes at Obama during a feisty Senate floor speech late Thursday in which he apologized to U.S. families.

“I can only imagine the anger and disgust they have at witnessing a broken government and a president and members of Congress who can’t seem to even agree sometimes on what day it is — let alone how to solve our nation’s debt crisis,” Manchin said.

The long-term political impact on Obama depends on a lot of factors still unresolved — such as what the final deal looks like, how future debates shape up on entitlements and taxes, and whether the economy starts to pep up.

The White House signaled confidence Friday that Congress would strike a deal by the Tuesday default deadline. On Wednesday, Carney said, Obama would travel to Chicago for campaign fundraisers.