The three Republican members of the City Council may belong to the same political party as Mayor Bloomberg, but their relationship with him is contentious and they have not yet committed to supporting him in November’s election.

The situation hit its low in January 2003 when James Oddo, the council’s minority leader, and a fellow GOP member, Andrew Lanza, stormed out of a Gracie Mansion dinner after Mr. Bloomberg praised the “courage” of Democrats who had passed his 18.5% increase in property taxes – a hike the Republicans adamantly opposed.

Now, however, Mr. Bloomberg is preparing for a tough campaign that will require strong support from the areas of Staten Island and Queens that the minority members represent. And the council Republicans – Mr. Oddo, Mr. Lanza, and Dennis Gallagher – don’t necessarily have a Republican, with a real shot of knocking out the mayor, to rally around.

“The problem,” a political science professor at Columbia University, Steven Cohen, said, “is that Bloomberg has to appear to not be too overly Republican in a Democratic city, but also has to make sure the Republicans in this city support him. He has to make sure his base is intact.”

Mr. Oddo has not sought to conceal his icy relationship with the administration. Though there was a thawing after the Gracie Mansion dinner, most notably when the two sides cooperated in creating a task force to address overdevelopment problems on Staten Island, there has never been a we’re-on-the-same-team spirit.

Mr. Oddo told The New York Sun yesterday that the administration called him several times last week, presumably to begin a warming of relations before the campaign. The minority leader has said he “hopes to support” Mr. Bloomberg but has left the door open. He and his small caucus could opt against endorsing anyone, or could endorse the mayor but not campaign aggressively for him.

“Why they never saw the value of embracing the three Republican council members I don’t know,” Mr. Oddo, during a phone interview, said of the Bloomberg team. “If they had, they’d have three pit bulls defending them on the floor of the council, in the media, and, perhaps most importantly, at home in the district.”

When asked on New York 1 at the end of last month about his relationship with the mayor, Mr. Oddo quickly responded: “Well, I didn’t send Ed Skyler a holiday card, if that’s what you mean,” referring to the mayor’s press secretary.

“When I campaigned for Mike Bloomberg in the heat, back then in that summer, I envisioned the type of relationship that Tom Ognibene, my predecessor, the former minority leader, had with Rudy Giuliani,” Mr. Oddo said during the broadcast. “He was a foot soldier in that revolution. He was a point person on the floor of the council…. That’s what I envisioned. … Three years later, that really never materialized.”

The relationship that did materialize is one in which the GOP council members are not hesitant to disagree publicly with the administration and are not privy to the administration’s day-today public-relations strategies.

Mr. Oddo even attended two news conferences last month with the Democratic speaker of the council, Gifford Miller, calling on the administration to find money for the hiring of additional police officers and school nurses. Mr. Miller is hoping to unseat Mr. Bloomberg in the mayoral election.

Even when the Republican mayor and the Republican council members share positions, relations are not always smooth. All three Republicans on the 51-member council, for example, voted against a recent bill to increase the cap on public campaign matching funds to $6 for every dollar privately raised, from the 5-to-1 formula that was on the books. The administration also opposed the bill. Mr. Oddo explained his vote at a hearing by saying despite his objection to “rich folks who spend an obscene amount of money” on campaigns, he could not justify that use of public money. Whether it was meant as a dig at Mr. Bloomberg, who spent $74 million of his personal fortune on the 2001 campaign, is unclear. The comment, however, would probably not have passed Mr. Oddo’s lips if the two had been closer partners.

Mr. Oddo described his relationship with the administration as “liberating” and said the mayor has been “respectful.” He said he has no problem siding with the mayor, as he did when he backed the mayor’s controversial smoking ban, but he can disagree just as easily.

The mayor’s spokesman, Mr. Skyler, said yesterday: “They are not always going to agree on everything, but the mayor respects them and is always open to strengthening their personal relationships.”

The mayor has a tall task in the election. Democrats, trying to capitalize on their huge edge in enrollment, are trying to align Mr. Bloomberg with President Bush. Republicans say he’s distanced himself by failing to employ high-ranking members of the party in his administration and by pushing a liberal agenda marked by his tax hike.

As a result, the mayor, who joined the GOP to run in 2001, is toeing a fine line. He needs a strong showing in the outer-borough, conservative pockets of the city.

“These are absolute crucial areas for the elections,” Mr. Gallagher, who represents Queens, said. “These are your swing districts. You have a lot of Reagan Democrats, Giuliani Democrats.”

Though the members of the council’s Republican caucus said it was too early for endorsements, political consultants predicted that the trio would be in lockstep with the Republican Party at the end of the day – even it was a marriage of convenience.

“My guess is that all of the Republican officials will fall in line with their ticket,” a political consultant, Allen Capelli, said. “I would be willing to make a fairly substantial wager on it, despite whatever dance is going on now.”

“I don’t think Bloomberg is going to have a problem on Staten Island,” Mr. Capelli said. “The mayor has good relationships with Guy Molinari, who is a bit of a local icon, and with the borough president, James Molinaro.”

Even in the smallest borough, where Mr. Bloomberg won more than three fourths of the vote in 2001,the mayor has taken small steps toward reconciling. He recently appointed two Republicans to judicial posts at Staten Island, the first two Republicans since he took office.