If Mayor Bloomberg decides to run for governor in 2010, Governor Paterson would confront him head-on despite what would seem to be an uneven matchup: a contest between a nationally known billionaire businessman with strong approval ratings and a former state legislator who is best known as Governor Spitzer’s emergency replacement.

Following press reports that Mr. Bloomberg is considering a gubernatorial bid after he completes his second term as mayor next year, the governor and his political team discussed the potential threat posed by the mayor and concluded that Mr. Paterson would not drop out of a race against the mayor, according to a source close to the state leader.

“If he did it, we’re going to run right at him,” a source close to Mr. Paterson said.

The term-limited mayor, who famously toyed with the idea of running for president, has not publicly expressed any interest in the state’s top job and has asserted his confidence in Mr. Paterson’s abilities.

Speculation about his political plans, however, surged last week after The New York Sun reported that Mr. Bloomberg has taken an interest in overthrowing the chairman of the state Republican Party and replacing him with a political ally, a move that would help the mayor forge an alliance with the party should he launch a campaign for governor as an independent or a Republican.

The New York Times reported Mr. Bloomberg had commissioned a poll to determine if voters would welcome him as their governor or would be open to overturning the city’s term limits law so he could extend his mayoralty another four years.

As the case of Eliot Spitzer demonstrated, the fortunes of a politician are subject to rapid reversals. Still, Mr. Bloomberg’s advantages against Mr. Paterson are overwhelming.

The mayor could drop $300 million of his fortune on a campaign — about five times the amount that Mr. Spitzer raised in the 2006 race. Mr. Paterson would likely have a hard time raising one-tenth of what Mr. Bloomberg would spend.

With an approval rating hovering around 75%, Mr. Bloomberg is in a strong position to claim most of the New York City vote, the largest bloc in the state and one that has historically been a decisive base of support for Democratic candidates for governor.

The bipartisan appeal of Mr. Bloomberg, an independent who has a close relationship with Albany Republicans, may deter a Republican from challenging him, allowing him to lock up upstate and suburban voters.

A May 19 Siena Research Institute Poll of registered New York voters found that Mr. Bloomberg, running as a Republican, would defeat Mr. Paterson in a theoretical matchup for governor. Forty-five percent of respondents said they would favor the mayor, compared to 36% for Mr. Paterson, with 20% saying they didn’t know whom they would prefer.

Almost three-quarters of New York City voters think Mr. Bloomberg would make a good governor, according to a Quinnipiac University poll taken in March.

The more relevant question is whether would Mr. Bloomberg want to be governor. His closest aides say his experiences in dealing (and more often than not pleading) with state lawmakers, who pride themselves on thwarting his will, have exasperated a mayor who has exercised dominion over the City Council.

“He would be frustrated on a daily basis with that job,” a Democratic assemblyman said. “Eliot Spitzer proved that the governor of New York does not have dictatorial powers. It is a position from which one must negotiate and deal with two truly independent bodies.”

Mr. Bloomberg may find the job of governor, whose responsibilities are more amorphous and whose impact on constituents harder to measure than that of a mayor, less rewarding.

“He would need to see a challenge that he could apply himself to and make a difference for people. I’m not sure he’s had that discussion with himself,” a former top aide to Mr. Bloomberg, William Cunningham, said.

Mr. Paterson, a former lieutenant governor and state Senator who has spent his adult life in Harlem, has yet to make a deep impression in his less than three months in office.

He has a reputation as a savvy political operator. What is unknown is whether his easygoing and affable governing style will translate into tangible legislative results.

The governor has begun to define his goals. He has vowed to impose discipline on the state’s finances and has proposed a cap on local property taxes.

Albany observers say the strengths and weaknesses of his administration will become more evident next year when he has the chance to craft his first budget and execute his agenda.

“We don’t know what kind of governor Paterson will be,” a professor of public policy at Baruch College, Douglas Muzzio, said.