Alvarez went 12-7 with a 2.65 ERA last year and made the All-Star team.
The team initially announced the right-hander had inflammation in his throwing shoulder. Manager Mike Redmond said there was the same problem with the elbow.
"It's definitely a concern anytime you have one of your star pitchers go down and have something sore," Redmond said. "We'll just have to monitor him as we go. ... It's tough not having him, for sure."
Alvarez (0-2) missed the first half of the 2013 season with shoulder inflammation. He had a stint on the disabled list last August with a shoulder issue.
Asked about a timetable for Alvarez, Redmond said "That's kind of the unknown right now."
Redmond said the initial plan is "calming down the inflammation and get him feeling good and healthy. How long it will be, I don't know.
"We're going to make sure he feels good and confident before he goes out and pitches again."
Alvarez gave up four runs in five innings and lost Sunday to Tampa Bay.
"It just looked like he was having a tough time getting the ball down in the zone," Redmond said. "He even dropped down and tried some different arm slots in that last inning, which sort of alerted me."
Redmond said Alvarez wanted to "keep pitching" and avoid the disabled list.
"The smart thing to do was to get him checked out and get him calmed down," he said, adding the goal is to make sure Alvarez "can finish the season strong."
The Marlins are already without ace Jose Fernandez, who is still recovering from Tommy John surgery and unlikely to return before mid-July.
Also, right-hander David Phelps was placed on the paternity list after his wife gave birth. He is expected to rejoin the team Thursday. Redmond said Phelps is the likely choice to replace Alvarez in the rotation on Friday.
Right-hander Jose Urena was recalled from Triple-A New Orleans to replace Phelps. Urena was expected to join the team by the start of Monday night's game at Atlanta.