Harvey was tagged Thursday night for six runs in six innings of a 12-4 loss to the Braves. After the game, the former ace took a firm stand against a possible move to the bullpen.
Asked Friday about Harvey's future, Callaway said, "I think he made that pretty clear yesterday, that he wants to be a starter, and I don't blame him. I think everyone wants to start. What we are trying to do as a team is do what's best for the team and for the players."
Harvey is 0-2 with a 6.00 ERA in four starts this year.
"I'm a starting pitcher," Harvey said after Thursday's game. "I've always been a starting pitcher. That's my mindset."
Callaway said he would like for a player to support a new role. He said that agreement is not necessary for a change to be made.
"Ideally you get buy-in," the first-year manager said. "But sometimes decisions are decisions, and people have to live with them. We'll do the best we can to help each and every one of these players succeed in whatever role they're in."
The decision on Harvey won't be made by Callaway alone. He said "there's been more talks" about Harvey's future and added that the Mets are "still not exactly sure what route we'll go."
Harvey has allowed a combined 14 runs in his past three starts. He was 5-7 with a 6.70 ERA in 2017.
The Mets must soon create room in the rotation for left-hander Jason Vargas, currently on the disabled list with a broken right hand. Vargas will make a rehab start on Monday in Las Vegas, instead of Sunday as originally planned.
Vargas tied for the major league lead in wins with 18 last season for the Kansas City Royals. His glove hand was broken by a comebacker in spring training.
"He will need to be evaluated after his session by the doctor to make sure he's good to go," Callaway said of Vargas.
Harvey's next scheduled start is Wednesday at the St. Louis Cardinals.
"If Vargas were still going to be on the DL, Matt would definitely start," Callaway said.
Harvey, 29, was an All-Star in 2013, then he missed the 2014 season following Tommy John surgery. He helped the Mets reach the World Series in 2015, then missed half of the 2016 season following surgery to correct thoracic outlet syndrome. He was out more than two months last season with a right shoulder injury.
"It's not a health thing anymore," Harvey said after Thursday night's loss.
Callaway said it is "hard to pinpoint" the reason for Harvey's struggles. He said it's not about bad mechanics.
"I think you look at his stuff and it's just down, his spin rate and all that stuff is down, the swing and miss," Callaway said.
Callaway could move another starter to the bullpen -- or to the minors -- to clear a spot for Vargas.
"We're looking at who could handle a bullpen role best," he said. "We're looking at does it make sense for somebody to go to Triple-A.
"There are a ton of things we're considering, and we're trying to leave no stone unturned right now."