The team says Wright was examined Tuesday at the Hospital for Special Surgery in New York after feeling discomfort. He will be re-evaluated after the two weeks and could be a designated hitter then.
"What's happening here is that the muscles around the shoulder have not re-engaged since the surgery," Alderson said. "That's taking more time than anticipated."
Wright, the team's captain, is owed $67 million over the next four seasons. He had surgery last June to repair a herniated disk in his neck.
His status for Opening Day remains questionable. If he cannot start the season, Jose Reyes, who started 50 games at third base in 2016, would likely start there.
"This is all part of the process of rehabilitating from the neck surgery," Alderson said. "It's taking longer than I'm sure David would have hoped, and we would have hoped, but it's part of the process."
Reyes said he would be ready if called upon to play third base.
"Third base is not a problem," Reyes said. "I'm ready. Wherever Skip [manager Terry Collins] wants to put me, whatever position in the lineup, I'm ready to do it."
Collins tried to stay positive about Wright, noting his previous ability to return from injury.
"Every time [Wright] comes back from an injury, you see him do something that people just rave about. I just think that if anybody can come back, he's probably a guy who can do it," Collins said. "But we'll have to wait and see how he progresses."
Wright, 34, a seven-time All-Star, has been limited to 75 regular-season games over the past two seasons. He was on the disabled list from April 15 to Aug. 24, 2015, when he strained his right hamstring and then developed spinal stenosis.
He returned and helped the Mets win their first National League pennant since 2000.
Last season, he was batting .226 with seven homers, 14 RBIs and 55 strikeouts in 137 at-bats through May 27 when he was sidelined by neck pain.
The Associated Press contributed to this report.