This time last year, the city comptroller, William Thompson Jr., said he had to disagree fundamentally with the direction the city was headed and Mayor Bloomberg’s poll ratings had to be in the basement for the Democrat to enter the 2005 mayoral contest.

“If I think I can beat him, I’ll run,” Mr. Thompson told The New York Sun.

Since then, Mr. Bloomberg’s poll ratings have been hovering in the high 40s and are rising, and it would be harder to make the case that the city is heading in the wrong direction. Mr. Thompson disagrees with the mayor on the West Side development project and his marketing agreements for the city. Nevertheless, crime is at historic lows, the local economy is boasting thousands of new jobs, and New Yorkers may not love this mayor but generally seem satisfied with the way he runs the city.

Mr. Thompson is expected to have a “final conversation” with his political advisers this week and announce his intentions next week. His aides declined to provide details. While the criteria he set out months ago have not been met, they said, Mr. Thompson still needs “a gut check” before making his final decision. Those close to him predict, however, that he will say he isn’t going to run for mayor and will seek re-election instead.

“He’d be insane to run,” a professor of politics at Baruch College, Douglas Muzzio, said. “He’d have to have a primary and it would be expensive and bruising. And then he would have to run against a mayor with a decent record and unlimited resources. There is no up-side for Thompson.”

Mr. Thompson, the son of a powerful Brooklyn judge, rose through the Brooklyn political ranks, did a tour on Wall Street, headed the Board of Education, and then, in his first run for elected office, landed the comptroller’s job in 2001.

Mr. Bloomberg’s political strategists are calculating that Mr. Thompson doesn’t have the stomach for a hard-fought primary and would rather wait this mayoral cycle out and take advantage of an easy run for a second term as comptroller. That’s why the Bloomberg campaign is positioning itself to go toe-to-toe with the former Bronx borough president who sought the mayoralty in 2001, Fernando Ferrer.

With Mr. Thompson out of the mix, Mr. Bloomberg would have two other possible African-American challengers: the Manhattan borough president, C. Virginia Fields, and a City Council member from Brooklyn, Charles Barron. While both seem capable of winning a good share of the black vote, neither has the kind of crossover appeal that analysts say Mr. Thompson possesses. Mr. Thompson is also the only one of the three who has won a citywide office.

Last week, Mr. Bloomberg seemed intent on wooing black voters, who, at this point, tend to regard him with skepticism. He visited a black congregation in Brooklyn on Sunday. On Tuesday, he announced that the Dance Theater of Harlem would reopen, thanks to an anonymous $500,000 contribution – that, it turned out, came from Mr. Bloomberg himself. The week ended with a $129 million high school reorganization in the Bronx and a breakfast at the International House of Pancakes in Harlem, where, on a bill of $81, Mr. Bloomberg left a $30 tip.

The mayor and his aides are keenly aware of the lukewarm support he generates in the black community. He has a 37% approval rating among black voters and a 46% disapproval rating, according to a poll last month by the Quinnipiac University Polling Institute. That was up from a 73% disapproval rating in July.

The trajectory of the numbers should provide some solace for Mr. Bloomberg, a Democratic political consultant, Hank Sheinkopf, said.

“What Bloomberg needs to worry about is the intensity of turnout in the black community,” he said. “Bloomberg is not the polarizing figure that Rudy Giuliani was. If anything, he has kept racial tensions calm and reduced them. He has fought for education, and that is a significant issue for blacks. People shouldn’t assume that Bloomberg can’t win the black vote.”

Three years ago, when the public advocate, Mark Green, ran against Mr. Ferrer, he alienated many black voters because his campaign was blamed for distribution in some white neighborhoods of an inflammatory cartoon depicting his opponent as a supplicant to the Rev. Al Sharpton. In the runoff vote that fall, more than 70% of black voters were said to have supported Mr. Ferrer, and in the general election, many black voters were thought to have cast their ballots for Mr. Bloomberg just to punish Mr. Green. If Mr. Bloomberg ends up running against Mr. Ferrer, he can’t count on those voters to return to his fold, the political analysts said.

Mr. Ferrer can’t take the black vote for granted either. Mr. Sharpton fired a salvo in Mr. Ferrer’s direction in the El Diario newspaper yesterday. In an article titled “Sharpton is not comfortable with Freddy,” using Mr. Ferrer’s nickname, Mr. Sharpton was quoted as saying Mr. Ferrer’s links to the Bronx Democratic Party would haunt him in next year’s race.

In particular, Mr. Sharpton singled out a former Bronx Democratic Party chairman, Robert Ramirez, who was Mr. Ferrer’s campaign manager in 2001. Mr. Sharpton said Mr. Ferrer needed to make changes in his campaign staff if he wanted to win this time around. “Freddy has to show new faces in the upper echelons of his campaign,” Mr. Sharpton was quoted as saying. “Freddy has to show his independence from the heads of the party, be they Latino, black, or white.”

That said, Mr. Sharpton did call Mr. Ferrer the “strongest candidate running.” He stopped short of endorsing him, however, and said Mr. Barron was his first choice.

“It isn’t going to be easy to beat Bloomberg,” Mr. Sheinkopf said. “In the 35 years I have been doing this, no incumbent has ever been defeated by a positive argument. They are defeated by negative comparative arguments. Thus far it is going to be difficult.”

At this early stage, Mr. Ferrer is testing themes with which he can hit Mr. Bloomberg during the campaign. The latest: that he is a man of the people, and the billionaire mayor is not.

“That isn’t going to wash,” Mr. Sheinkopf said. “There has to be a specific dynamic that gets people to move out against an incumbent, and I don’t see that materializing right now.”

Jennifer Bluestein, a spokeswoman for the Ferrer campaign, declined to comment yesterday.

The other likely Democratic mayoral candidates are the council speaker, Gifford Miller, and a congressman from Brooklyn and Queens, Anthony Weiner.