CHICAGO (Reuters) - U.S. Sen. Hillary Clinton left the door open to becoming secretary of state on Friday, a day after a meeting with President-elect Barack Obama at which a Democratic official said they discussed the job.

Obama’s selection of Clinton would be a bold move that would bring into his coming administration a high-profile former rival who had questioned his level of experience and his foreign policy ideas in the Democratic primary battle earlier this year.

In introductory remarks at a speech in Albany, New York, Clinton, wife of former President Bill Clinton, did not confirm or deny that she had a secret meeting with Obama on Thursday. Democratic officials said the meeting did take place.

“I’m not going to speculate or address anything about the president-elect’s incoming administration. And I am going to respect his process and any inquiries should be directed to his transition team,” Clinton said.

The Obama’s transition office in his hometown Chicago refused to comment.

The Washington Post, meanwhile, said Obama met on Friday with another former rival, New Mexico Gov. Bill Richardson, and that they talked about the secretary of state position.

A Democratic official told Reuters that Clinton and Obama “had a serious meeting but the question was whether an actual offer was made.”

CNN quoted Democratic sources as saying Obama and Clinton had a serious discussion to gauge her interest in becoming secretary of state and that she left the meeting with the impression the job was hers if she wanted it.

Nominating Clinton for the job would be a dramatic step by Obama as he seeks to build a high-powered team in Washington. He defeated the New York senator in a sometimes bitter duel for the presidential nomination, then miffed her supporters by not choosing her as his vice presidential running mate.

Obama also scheduled a meeting in Chicago on Monday with Republican Sen. John McCain, their first face-to-face talks since Obama defeated him in the presidential election on November 4.

Obama spokeswoman Stephanie Cutter said Obama and McCain would discuss ways to bring about a “more effective and efficient federal government.” McCain had campaigned on reforming government by cutting out programs and reducing spending.

WOULD CLINTON TAKE THE JOB?

Washington was abuzz with the news that Clinton might be on Obama’s short list and be a member of a “team of rivals” in much the same way Abraham Lincoln consolidated power by bringing his opponents into his Cabinet.

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Many experts thought she would be a strong choice for secretary of state -- she would be the third woman to hold the job after Madeleine Albright and Condoleezza Rice.

“You need someone who can be taken seriously internationally, who has that kind of experience and policy ability, but also has the gravitas to make people sit down at the table,” Democratic strategist Liz Chadderdon said.

But the question was whether she would take the job if offered. An alternative could be to remain in the Senate and use her clout to have a say in Obama’s proposals and keep an independent platform for another possible run for president.

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There were also questions about how her spotlight-loving husband would be managed if his wife were the top diplomat.

Obama has already turned to many former members of Bill Clinton’s administration for help in his transition to power, including his chief of staff, Rahm Emanuel, and co-chairman of his transition team, John Podesta.

Considering Clinton for secretary of state would mean Obama was expanding his search beyond other candidates mentioned for the job, such as Massachusetts Sen. John Kerry, a Democrat who lost the 2004 presidential election to George W. Bush; Nebraska Sen. Chuck Hagel, a Republican who backed Obama over McCain this year; and Richardson, who was the U.S. ambassador to the United Nations during Bill Clinton’s presidency.

Obama has been closeted all week in meetings in his transition office in Chicago as he prepares to take over the presidency on January 20.

New Jersey Democratic Gov. Jon Corzine said on MSNBC that Clinton would be great as secretary of state.

“She probably knows every major foreign leader. There’s already a relationship where she can sit and talk directly about the problems that exist either on a bilateral or multilateral basis,” he said.

Corzine would not address speculation that he is a candidate to be Obama’s Treasury secretary, amid reports that another candidate for the position, Lawrence Summers, was fading due to opposition from women’s groups upset at him during his tenure as president of Harvard University.