Could this be the beginning of a beautiful friendship?

Mayor Bloomberg invited the president of the United Federation of Teachers, Randi Weingarten, to sit in his box Tuesday and root for the Yankees in the first game in the American League Championship Series against the Boston Red Sox, setting off speculation that there is a thaw in their often-contentious negotiations over a new contract for New York City teachers.

“Randi has been outside the normal give-and-take of being on opposite sides of the negotiating table, she has been a friend of mine since I came into public service,” Mr. Bloomberg told reporters yesterday. “She’s a very smart woman and has a lot of interests, and we don’t talk about the contracts when we’re there, there are lots of other things to talk about in the world.”

Mr. Bloomberg and Ms. Weingarten have locked horns over the teachers contract for more than a year now. The mayor wants teachers to provide some productivity enhancements in exchange for a raise. Ms. Weingarten has said teachers are stretched too thin as it is. She has been holding out for a better deal than the one Mr. Bloomberg forged with a major union of city workers, District Council 37, which had workers giving up comp time and some vacation days to get more money.

While Mr. Bloomberg sought to play down the baseball-game invitation, those close to the teacher contract talks say there has been a noticeable change in the tenor of the discussions. The city put a short contract on the table several months ago, which teachers thought was an outrage. That contract has since been withdrawn from the discussions. In addition, those close to the talks said the two sides formerly were at cross-purposes and met only to talk past each other, but now apparently are actually talking about the same issues, even if they are arguing about them.

“I’m sure he wants to send a signal we’re talking,” Ms. Weingarten told The New York Sun in an interview.

She also said, however, that she was more of a Mets fan than a Yankee supporter.

Some political analysts say they think the mayor is determined to forge a teachers contract before November 2005, when he wants to win re-election. Certainly the change in tone within the talks suggests that something is different.

Mr. Bloomberg tried to play down such speculation, telling reporters that his younger daughter, Georgina, had also talked to Ms. Weingarten during the game.

“I said to my younger daughter, ‘What did you talk about,’ and she said, ‘Things,’ ” Mr. Bloomberg said. “I can guarantee you that Georgina Bloomberg did not talk about the UFT contract.”