Joel Segal, the agent for the defensive back, has been in contact with a number of teams. Fitzpatrick, the 11th overall pick in the 2018 draft, is unhappy with playing multiple positions, sources said.
Teams initially have been resistant to the Dolphins' apparent asking price that includes a first-round pick, sources said.
"I can't focus on this trade, and this and that," Fitzpatrick told reporters Friday. "I'm focused on Sunday. I've got to focus on the New England Patriots."
Dolphins coach Brian Flores said Friday that he anticipates Fitzpatrick being a big part of Sunday's game plan against the Patriots.
Flores declined to comment whether Fitzpatrick will remain with the team through the season, but he said he doesn't think it's unusual for a player to stay with a team despite not wanting to.
Dolphins general manager Chris Grier was unavailable for comment late Thursday night, as was Segal.
A Dolphins official told ESPN they are not actively shopping Fitzpatrick, but did not deny that they have given permission to Segal to discuss a trade with teams.
Fitzpatrick, who was projected as a cornerback/safety out of Alabama, played three or four different positions against Baltimore. He has been uncomfortable with his constantly fluctuating role on the Dolphins' defense throughout the offseason.
During Miami's joint practices with the Tampa Bay Buccaneers in August, Fitzpatrick's mother, Melissa, mentioned on Twitter that he was being incorrectly used at strong safety to suit other players' skill sets.
Fitzpatrick confirmed that he agreed with his mother's thoughts after that practice, saying, "She feels very strongly. She's not wrong. Coach has asked me to do something right now. I got to do what they ask me to do. If we have to have some discussions in the future, we'll have those discussions.
"I'm not 215 pounds, 220 pounds. So playing in the box isn't best suited for me, but that's what Coach is asking me to do."
A personnel man for one team had praise for Fitzpatrick as the "ideal slot corner who can cover, blitz off the edge, [is] intelligent, a good tackler and a good special-teams player," but doubted that teams would be willing to part with a first-round draft choice in 2020.
Another GM felt that the Dolphins were unrealistic about their expectations on compensation for Fitzpatrick after "they got an arm and a leg for [Laremy] Tunsil from the Texans" before Week 1.
ESPN's Cameron Wolfe contributed to this report.