The latest update on Teixeira continues what has been a calamitous spring training for the injury-plagued Yankees.
New York already will be without slugging center fielder Curtis Granderson, who is slated to miss at least all of April with a fractured forearm, and star third baseman Alex Rodriguez, whose hip surgery will keep him sidelined until at least after the All-Star break.
Starting pitcher Phil Hughes is rehabbing from a bulging disk in his back and may not be ready for the beginning of the season.
Manager Joe Girardi said the Yankees have to "find a way" to get through these injuries, but the organization realizes this is not the way to start a season.
"This ain't good," said Yankees general manager Brian Cashman, who was sitting in a wheelchair because of a fractured fibula and dislocated ankle from a sky-diving accident Monday.
Cashman gave the news after the Yankees lost 8-2 to the Dominican Republic World Baseball Classic team on Wednesday.
Teixeira was hurt swinging a bat Tuesday in an indoor cage during pregame warmups with the United States' World Baseball Classic team. Initially expected to miss just eight-to-10 days, Teixeira was examined in New York by team doctors, who said he has a strained ECU tendon in his wrist.
Teixeira has been told not to do anything for the next four weeks. If he heals according to schedule, he will need another four to six weeks to be ready to play in the majors.
The Yankees' backup first basemen include Juan Rivera, Dan Johnson and Kevin Youkilis, who already is replacing Rodriguez at third base. Cashman and Girardi discussed Teixeira's injury almost immediately after learning about the diagnosis, so they did not yet have a plan for what the Yankees will do at first base.
One option in play is moving Youkilis from third to first. Upon the the initial 10-day diagnosis Tuesday, Girardi already had told Youkilis to start taking some grounders at first.
"It was mentioned in passing," Girardi said.
If Youkilis is needed extensively at first base, the Yankees could turn to Eduardo Nunez or Jayson Nix as their third baseman. New York previously had been hesitant to move Nunez, who will back up shortstop Derek Jeter.
Jeter and star closer Mariano Rivera are coming back from season-ending injuries in 2012. Both veterans could make their spring debuts this weekend.
Teixeira, who will turn 33 in April, dealt with a respiratory infection and a calf problem in 2012 that limited him to 123 games. He finished the season with a .251 average, 24 home runs and 84 RBIs.
Cashman will look outside of the Yankees' camp to replace the injured stars, but he was not optimistic about making a major deal.
"This is not the time of year to make any significant moves," Cashman said. "Usually, movement takes place after the draft unless people are trying to cut garbage."
The Yankees brought in former first baseman Travis Hafner this offseason, but see him exclusively as a designated hitter.
"He is a field-goal kicker," Cashman said. "I know what I got. I want him to be that. But he hasn't picked up a glove in seven years. I'm not making fun of him. He is amazing hitter, but he has a history of injuries and we are not going to put him in a position by trying to put a glove on him and get him hurt. He is our DH."
The Yankees' Opening-Day roster will look much different than a year ago, when they led the majors in homers. As of Wednesday, the only player who hit 20 or more homers in 2012 that will be in this year's initial lineup is Robinson Cano.
Information from The Associated Press was used in this report.