Premier League big boys Manchester United and Everton fancy snapping up the 18-year-old, who is yet to make a senior appearance for Bolton.
But, according to The Sun on Sunday, they have been stunned by the Trotters’ demands.
And it’s claimed they have been put off from firming up their interest.
Turner is rated as one of the country’s hottest goalkeeping prospects.
The Wilmslow-born keeper has represented England at Under-19 and Under-18 level.
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He is currently on loan with non-league side Stalybridge Celtic until the end of January.
Bolton, who won promotion from League One last term, have Turner under contract until June 2019.
He was a regular on Phil Parkinson’s bench last season and was named the club’s young player of the year.
Meanwhile, Bolton exited the FA Cup yesterday after losing at home to Premier League side Huddersfield.
Terriers duo Rajiv van la Parra and Danny Williams scored within a minute of each other after the break.
Derik Osede gave Bolton hope with a header but they could not find the second goal to force a replay.