The England international missed training on Friday after calling in sick, heightening speculation that he will soon be moving on.
Portsmouth executive chairman Peter Storrie said on Thursday that Defoe had talked about moving on. He has been linked with a return to Tottenham Hotspur to rejoin former Portsmouth manager Harry Redknapp, who was also his boss at West Ham United.
"He has a cold and the doctor is on the way to see him," Adams told Sky Sports News.
"I don't disbelieve anyone, I trust him and have a good relationship with him."
Defoe is unlikely to be considered for Saturday's home game against Bristol City in the first game of Portsmouth's FA Cup defence and Adams was circumspect when asked how he felt about Defoe's desire to leave.
"I'm not pleased because I want people to play for Portsmouth," he said.
"But it's better knowing ... if he's got the honesty to come in and tell you. I'm trying to find a positive as he's a fantastic player and one I don't want to lose.
"He's annoyed that I tactically changed the team and didn't play him against Liverpool and Arsenal but I told him my reasons for it."
Asked if he thought Defoe would play for Portsmouth again, Adams said: "I don't know."
