Premier League sides Brighton & Hove Albion and Crystal Palace have been most strongly linked with the 21-year-old, who has gone six games without finding the net.
Dembele was kept on the bench for Celtic's 6-0 win over Brechin in the Scottish Cup on Saturday afternoon, and Rodgers admits that off-field talk is having a negative effect on the young French striker.
"There has been nothing official as yet. I know some enquiries have come through to Peter [Lawwell, chief executive] but nothing concrete," he told reporters after the comfortable victory at Celtic Park.
"The boy is fine, he is training, he is working but it is clear that he has probably not been quite the same as what he has been for probably over a year or so. My job as the manager is to ensure that the focus is purely on Celtic and the team.
"He is a good kid and for a lot of players January is a difficult window - lots of speculation - but my job is always to protect the club and make sure that everyone who plays is totally focused. We will see what happens between now and the end of the month and will take it from there."
Dembele has eight goals in all this season, most recently netting in the 2-0 win over Motherwell on November 26.