Bristol City and Southampton are also interested in the midfielder, who won the Scottish Football Writers' Association's Young Player of the Year award.
Turnbull has two years left on his contract but his performances for the Steelmen have caught the eye.
Reports in the Daily Record, suggest the Fir Park side are looking for a transfer fee comprising an upfront payment of £2.5 million, with a further £500,000 in add-ons.
'Well chief executive Alan Burrows tweeted on Friday: "Getting some questions re David Turnbull, given today’s newspaper report.
"I try not to reply to speculation (for obvious reasons), but to reiterate what I’ve already gone on public record as saying.
"We will only consider offers of significantly more than our record fee received."
Motherwell's record fee is £1.75 million, which Celtic paid for Phil O'Donnell in 1994.