WASHINGTON — Mayor Giuliani’s past support for gay rights may put him at odds with social conservatives, but it is in tune with one small-yet-vocal constituency: the Log Cabin Republicans — a gay GOP group with a history of complicating the nomination paths for the party’s presidential contenders.

The Log Cabin group claims a membership of about 20,000 gay and lesbian Republicans nationwide and has fought against a federal amendment banning same-sex marriage.

Its members have long supported Mr. Giuliani, who as mayor of New York signed a landmark domestic partnership bill that guaranteed the same benefits to gay and lesbian couples as those given to heterosexuals. The Log Cabin Web site lists Mr. Giuliani as a model for its brand of Republicanism, and the former mayor spoke to the organization’s national convention in 1999. That same year, the group’s New York branch contributed $1,000 early in his campaign for Senate, which he called off to focus on getting treatments for prostate cancer.

Whether he will seek their support this time around, however, is another question. Republican candidates have kept the group at a distance in recent elections, owing to the clout that social conservatives have shown at the polls. In 1995, Bob Dole’s campaign returned a contribution from the Log Cabin group while in the heat of a primary battle. The move was seen as an attempt to mollify the right wing of the party, and months later, Mr. Dole reversed course and said it was a mistake by his staff.

In 2000, many members gave vocal support and financial backing to Senator McCain, and some later felt snubbed by President Bush, even after he met with gay leaders upon securing the Republican nomination. Mr. Bush declined to meet with the group during the primaries against Mr. McCain. Log Cabin made no endorsement in 2004, and its president, Patrick Sammon, said they have been “disappointed” with Mr. Bush.

Mr. Giuliani appears to have no qualms about accepting an endorsement or contribution from the Log Cabin Republicans if it is offered. “Mayor Giuliani is reaching out to all Republicans across the country,” a campaign spokeswoman, Maria Comella, said in a statement. “We welcome support from across the Republican” spectrum.

Support from the Log Cabin Republicans likely wouldn’t help Mr. Giuliani get the party’s nomination, and it could hurt, a political science professor at the University of Virginia, Larry Sabato, said. “My guess is if Giuliani gets nominated, he will have had to have made some pledges to evangelical Christians along the way,” he said.

Yet Log Cabin’s official backing is far from assured, as the group says it is proceeding cautiously this year. In an interview, Mr. Sammon praised Mr. Giuliani’s “inclusive” record, but he said the group is not planning to issue an endorsement in the primary. “That obviously could change as the campaign goes on,” he said.

He said the group’s message is for candidates not to run on “divisive social issues” if they want to win the general election. It is a warning that the group thinks will carry more weight as large, socially moderate states such as California and New York plan to move up their primaries to early February.

Mr. Sammon would not say whether the organization has had contact with any of the Republican campaigns, but he said it would be reaching out to them as the year goes along. “With all the Republican candidates, it’s early in the campaign, and we’re going to be following the campaign closely,” Mr. Sammon said.

Mr. Giuliani supports civil unions, but not same-sex marriage. He has said he would likely oppose a constitutional amendment banning it, but he has not ruled out supporting it in the future.

Log Cabin has a history with each of the top three Republican candidates. While many members supported Mr. McCain in 2000, and the group endorsed him in 2004, Mr. Sammon suggested his record was mixed. He pointed out that the senator opposed a federal marriage amendment but supported a state version in Arizona. “I think it’s too soon to tell exactly how things are going to go with the direction of his campaign,” Mr. Sammon said.

A friendly relationship with a former Massachusetts governor, Mitt Romney, has turned sour. Mr. Romney has moved to the right on social issues in recent years, including gay rights. Log Cabin has circulated a letter that Mr. Romney sent to its Massachusetts branch during his campaign for Senate in 1994. In the letter, Mr. Romney thanks the group for its endorsement and asserts that he would “provide more effective leadership” on establishing full equality for gays and lesbians than his Democratic opponent, Senator Kennedy, who is often called the “liberal lion” of the Senate.

“I don’t think he has credibly explained how that shift has happened,” Mr. Sammon said. “It certainly looks as if it’s all about politics.”
The campaigns of Messrs. McCain and Romney did not respond to questions about the Log Cabin Republicans.