The 25-year-old has reportedly attracted interest from fellow Magners League sides Leinster, Munster and Ulster after establishing himself as one of the world's leading wingers thanks to a string of fine performances with the Lions.
Bowe started all three Tests against the Springboks this summer and was widely regarded as one of the most consistent performers on tour.
He played in six of the 10 tour matches, scoring four tries and switching to outside centre for the final Test victory in Johannesburg.
"I don't think anyone wants to lose a player of Tommy's quality," said Ospreys director of rugby Johnson.
"I don't know exactly the situation (about contract talks), but we are confident of keeping him.
"He's a quality player and where he has come from in the last 18 months has been outstanding.
"He is a valuable contributor to our team and our squad. We are privileged to have him."
Bowe joined the Ospreys from Ulster in the summer of 2008 and has been a key performer for the region ever since, playing in 18 Magners Leagues games, including seven of the Ospreys' eight fixtures so far this season.