Newly re-elected, President Barack Obama moved quickly Wednesday to open negotiations with Republican leaders over the main unfinished business of his term: a major deficit-reduction deal to avert the “fiscal cliff” that America could drive over.

Obama, who also faces Cabinet shakeups in his second term, called House Speaker John Boehner in what was described as a brief and cordial exchange on the need to reach some budget compromise in the lame-duck session of Congress starting next week. Later at the Capitol, Boehner responded with his most conciliatory offer to date on Republicans’ willingness to raise tax revenues, though not top rates, as part of a spending cut package.

“Mr. President, this is your moment,” said Boehner, whose party still holds a healthy House majority. “We’re ready to be led — not as Democrats or Republicans, but as Americans. We want you to lead, not as a liberal or a conservative, but as president of the United States of America.”

His statement came a few hours after Sen. Harry Reid, leader of the Democratic Senate majority, extended his own olive branch to the opposition, while also saying his party would not be pushed around. Reid, a former boxer, said, “It’s better to dance than to fight.”

Both men’s remarks followed Obama’s own overture in his victory speech Wednesday.

“In the coming weeks and months, I am looking forward to reaching out and working with leaders of both parties to meet the challenges we can only solve together: reducing our deficit, reforming our tax code, fixing our immigration system, freeing ourselves from foreign oil,” he said.

After his speech, Obama tried to call both Boehner and the Senate Republican leader, Mitch McConnell, but was told they were asleep.

The efforts from both sides, after a long and exhausting campaign, suggest the urgency of acting in the few weeks before roughly $700 billion in automatic tax increases and across-the-board spending cuts take effect at year’s end — the fiscal cliff. A failure to reach agreement could arrest any economic momentum.

TAXES ON WEALTHY

If Obama got a mandate for anything after a campaign in which he was vague on second-term prescriptions, he will claim one for his argument that wealthy Americans should pay more in income taxes. That stance was a staple of Obama’s campaign stump speeches for more than a year. A

Specifically, Obama has called for extending income tax cuts only for households with taxable income of less than $250,000 a year.

Boehner, in his public remarks Wednesday, sought to avoid a White House tax trap that would have Republicans boxed in as defenders of the wealthy.

Speaking for Republicans after a conference call with his congressional colleagues, Boehner said he was ready to accept a budget deal that raised federal revenues, but not the top rates on high incomes. And the deal, he said, also would have to overhaul both the tax code and programs such as Medicare and Medicaid, whose growth as the population ages is driving projections of unsustainable future debt.

Instead of allowing the top rates to go up, which Republicans say would harm the economy, Boehner said Washington should end some deductions and loopholes to raise revenues. The economic growth that would result from a significant deficit reduction compromise would bring in additional revenues as well, he said.

Corporate America and financial markets for months have been dreading the prospect of a partisan impasse. Stocks fell Wednesday, with the Standard & Poor’s 500 Index closing down 2.4 percent. The reasons for the drop were unclear, given that stock futures did not drop significantly Tuesday night, as the election results became clear. Analysts cited fears about the economic impact of such big federal spending cuts and tax increases, but also about new economic troubles in Europe.

While Obama enters the next fray with heightened leverage, the coming negotiations hold big risks for both parties and for the president’s ability to pursue other priorities in a new term, such as investments in education and research, and an overhaul of immigration law.

A NEW-LOOK CABINET

The president flew back to Washington from Chicago late Wednesday, his post-election relief reflected in a playful race up the steps of Air Force One with his younger daughter, Sasha. At the White House, he prepared to shake up his staff to help him tackle daunting economic and international challenges. He will study lists of candidates for various positions that a senior adviser, Pete Rouse, assembled in recent weeks.

The most prominent members of his Cabinet will leave soon. Secretary of State Hillary Rodham Clinton and Treasury Secretary Timothy Geithner long ago said that they would depart after this term whether Obama won re-election or not, and Defense Secretary Leon Panetta, previously the head of the CIA, has signaled that he wants to return to California sometime in the coming year.

Also expected to depart is David Plouffe, one of the president’s closest confidants.

Obama is expected to reshuffle both his inner circle and his economic team as he accommodates the changes. For example, Jacob J. Lew, current White House chief of staff and former budget director, is said to be a prime candidate to become Treasury secretary.

For the foreseeable future, the holder of that job is likely to be at the center of budget negotiations, and Lew has experience in such bargaining dating to his work as a senior adviser to congressional Democrats 30 years ago in bipartisan talks with President Ronald Reagan.

“They’ve been thinking about this for some time and they’re going to have a lot of positions to fill at the highest levels,” said former Sen. Tom Daschle, who has close ties to the White House.

Both Presidents Bill Clinton and George W. Bush ended up replacing about half of their Cabinet members between terms, and Obama could end up doing about the same, especially since his team has served through wars and economic crises.

John Podesta, a chief of staff for Clinton and Obama’s transition adviser, said, “There’s a certain amount of new energy you want to inject into any team.”

There is talk about bringing in Republicans and business executives to help rebuild bridges to both camps. The one Republican in the Cabinet now, Transportation Secretary Ray LaHood, has said he will leave. One possible candidate, advisers say, could be Sen. Olympia Snowe, a moderate Republican from Maine who is retiring.

A front-runner for secretary of state appears to be Sen. John Kerry, D-Mass., and Democrats claimed that worries about losing his Senate seat to the Republicans in a special election had diminished with Tuesday’s victories.

Attorney General Eric Holder, once expected to leave, now seems more likely to stay for a while. Secretary of Homeland Security Janet Napolitano would like to be attorney general and is widely respected in the White House.

Among other Cabinet officers who might leave are Ron Kirk, the trade representative; Steven Chu, the Energy secretary; Ken Salazar, the Interior secretary; Tom Vilsack, the Agriculture secretary; and Lisa Jackson, the Environmental Protection Agency chief.

Valerie Jarrett, the president’s longtime friend and senior adviser, plans to stay, according to Democrats close to her.

It might be weeks before Obama starts making personnel announcements, officials said.