(CNN) -- After eight years as senator from New York, Hillary Clinton is trading places, moving from Congress to the incoming administration.

Former President Bill Clinton has been mentioned as a possible replacement for his wife in the Senate.

On Monday, President-elect Barack Obama announced that he asked his former rival to be his secretary of state.

That means the scramble begins to replace Clinton on Capitol Hill. Among those mentioned to take her seat as New York's junior senator is her husband, former President Bill Clinton.

At a news conference in Chicago, Illinois, on Monday, after Obama announced her selection, Clinton said she wanted to "thank my fellow New Yorkers who have, for eight years, given me the joy of a job I love with the opportunity to work on issues I care deeply, in a state that I cherish."

Clinton added that "leaving the Senate is very difficult for me."

The task of choosing a successor falls to David Paterson, New York's Democratic governor. Whomever he picks would serve for two years, before a special election in November 2010 to decide who fills the last two years of Clinton's term.

Paterson has a strong bench to choose from. There are a number of contenders, including at least eight members of New York's delegation in the House of Representatives, New York State Attorney General Andrew Cuomo, Buffalo Mayor Byron Brown, Bronx Borough President Adolfo Carrion, Nassau County Executive Thomas Suozzi, Caroline Kennedy, and her cousin, Robert Kennedy Jr.

"This is not an election. This is not a campaign. It's a constituency of one. David Paterson. It's all about what the governor wants to do," said political analyst Stuart Rothenberg, editor and publisher of The Rothenberg Political Report.

"Paterson has said he would prefer someone from upstate New York, or a woman or an Hispanic candidate," Rothenberg said.

As for some of the more unconventional picks, Rothenberg said Paterson could "try to make a splash with a big name like Robert Kennedy Jr." or a "quirky interesting pick" of someone like Caroline Kennedy, who is not a politician.

But some are suggesting the former president should take his wife's seat.

In an op-ed column last week in The Washington Post, journalists Karl Meyer and Shareen Blair Brysac urged Paterson to "send Bill Clinton to the Senate."

If that happened, Clinton would become the third former president to go from the White House to Capitol Hill.

President John Quincy Adams lost his re-election bid in 1828. Two years later he returned to Washington after winning election as a congressman from his home state of Massachusetts. He served in the House of Representatives until his death in 1848.

President Andrew Johnson also served as a Senator from Tennessee in 1875, 7 years after the Senate acquitted him of impeachment charges. He died a few months after taking office.

Bill Clinton would bring gravitas to the job, and he obviously knows his way around Washington.

But some Democratic strategists who used to work for Bill Clinton don't think the former president would want to go from leader of the free world to being the junior senator from New York. Clinton's office deferred to Paterson's office when asked for a comment.

In a statement released Monday in which he praised the choice of Hillary Clinton for the nation's top diplomatic post, Paterson said, "In order to appoint the best possible candidate to replace Sen. Clinton, I am consulting with a wide variety of individuals from all across New York State.

"I expect to announce Sen. Clinton's replacement when the position becomes officially vacant," he said.

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