Wainwright, 41, who was scheduled to be the Cardinals' starter on Opening Day for the seventh time as he enters his 18th season with St. Louis, will begin the season on the injured list, Marmol said.
The injury happened during a weight room workout Tuesday with Team USA during the World Baseball Classic.
"We'll continue to evaluate over the next few days but no timetable at the moment," Marmol told reporters. "There is a spot in the rotation now open, and we're going to have to sit down and make sure that we're not missing anything that goes into that."
Wainwright has thrown 15 innings this spring -- seven in Grapefruit League action and another eight during the WBC. He last pitched Sunday in Team USA's semifinal victory against Cuba, allowing just one run in four innings while earning his second win in two starts in the tournament.
"I knew it wasn't great when it happened," Wainwright told reporters about the injury. "... It could be much worse. It's just a strain. We'll be out a few weeks.
"... It's disappointing. I wanted to start [Opening Day]. I wanted to pitch a full season this year, but I'd like a full postseason more than I'd like a full April."
He decided to return for another season with the Cardinals after going 11-12 with a 3.71 ERA in 32 starts in 2022. It was his first losing season in which he made at least 20 starts.
Wainwright, who has said that he would retire after the season, has gone 195-117 with a 3.38 ERA over 17 seasons with the Cardinals. He's a three-time All-Star who's posted two 20-win seasons, along with winning two Gold Gloves.
He needs five wins to reach 200 for his career. He is 15 behind Jesse Haines for the second-most wins in franchise history; Bob Gibson is the leader with 251.
Jake Woodford is the leading candidate to take over Wainwright's place in the starting rotation.
The Associated Press contributed to this report.