Injuries limited Alvarez, an All-Star in 2014, to four starts in '15, and he was 0-4 with a 6.45 ERA.
The Marlins are not expecting the right-hander to be ready for Opening Day. If the club was confident he could contribute for at least half the season, it may be willing to tender him a contract for 2016. But along with the shoulder, Alvarez also has had a partial tear in his right elbow that he has been pitching through for years.
Miami has until 11:59 p.m. ET on Wednesday to decide whether to tender all 10 of its arbitration-eligible players.
A tender means the team retains control of the player and would then work toward deciding on the salary for 2016. Alvarez made $4 million in '15, and his contract would be roughly the same if he is tendered.
Along with Alvarez, Phelps and Crow, the other seven Marlins eligible for arbitration are second baseman Dee Gordon, shortstop Adeiny Hechavarria, and right-handers Jose Fernandez, Tom Koehler, Carter Capps, A.J. Ramos and Bryan Morris.
Those seven are expected to be tendered. Miami also has opened preliminary talks with Gordon, the National League batting champion, on a multiyear contract.
Phelps, who appeared in 23 games and made 19 starts in 2015, had his season cut short in August due to a stress fracture in his right forearm. Offering versatility to start or relieve, he was 4-8 with a 4.50 ERA in 112 innings.
Crow, meanwhile, missed the entire season due to Tommy John surgery, which took place in early April. The 29-year-old was acquired from the Royals the day after Thanksgiving in 2014.
With Crow and Phelps, the organization must decide if they can overcome their injuries and be able to contribute at some point during the season.
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro and listen to his podcast. This story was not subject to the approval of Major League Baseball or its clubs.