Barnes has broken into the first-team fold at Home Park this season, recording seven Championship appearances to date.
The 19-year-old's current deal is due to expire in the summer and recent media reports speculated that Fulham, Stoke City and West Ham United are all eyeing a move for his services.
But Sturrock told The Herald: "I have taken Ashley into my first team and he has now played four or five games where he has been very competitive.
Rocket science"As a manager, I pick what I see - and I like what I see with Ashley. It doesn't take a rocket scientist to appreciate I'm going to be trying to do a new deal for him."
Sturrock continued: "It seems to be a coincidence that his agent has phoned me twice lately and told me, 'We will need to start thinking about a new contract'.
"What has probably happened is that his agent has been sat in the stands and, lo and behold, the scouts from Fulham, Stoke and West Ham have been there.
"All of a sudden, there are stories in the papers saying these teams have got an interest. When managers are sat in the stand, I will begin to worry."