The Yankees gave permission for Teixeira to stay in the New York area with his wife, Leigh, who gave birth to a baby boy, William Charles, on Tuesday.
Yankees manager Joe Girardi indicated that Teixeira will rejoin the club in Kansas City, although it was not definitive if that will be on Thursday, when the Bombers open a four-game series with the Royals at Kauffman Stadium.
"I expect to have him sometime in Kansas City," Girardi said. "We want to make sure everything is OK. I think [Teixeira's return] will be sometime in Kansas City."
With Teixeira absent, the Yankees started Lance Berkman at first base on Tuesday to open a brief two-game series with the Rangers.
Marcus Thames served as the designated hitter and Ramiro Pena filled in at second base for ailing Robinson Cano, who is suffering from a fever and cold.
Teixeira hit his 26th home run of the season in Monday's 2-1 loss to the Red Sox, completing a seven-game homestand during which he blasted five homers.
The only Major Leaguer to reach 80 runs scored and 80 RBIs so far this season, Teixeira has reached base in 53 of his last 55 games since June 6.
Young William Charles is the couple's third child, joining a son, Jack, and a daughter, Addison.
"We're all happy for him," Girardi said.
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.