The Parkhead side have approached the Steelmen with an official enquiry, according to the Daily Record.
Twelve months ago John McGinn was seen as a key target for Brendan Rodgers. However, the club dallied, allowing Aston Villa to swoop in and sign the midfield dynamo from Hibernian for £2.7million.
In hindsight the price was a bargain for the Birmingham club with the Scotland international playing a pivotal role in helping Villa win promotion to the Premier League via the Championship play-offs.
Celtic are moving quick for Turnbull - one of the rising stars of Scottish football - with reportedly nine other clubs showing interest, including Premier League sides Southampton and Burnley.
Earlier in the week it was reported that they were considering £1.2million, but Motherwell chief Alan Burrows took to Twitter to state that the club will only sell their wonderkid for a record sale, more than the £1.75million they received from Celtic for Phil O'Donnell.
The Steelmen want at least £3million for the 19-year-old who hit 15 league goals from midfield this season just past.
In January Turnbull, a boyhood Motherwell fan, extended his deal at Fir Park until 2021, putting the club in a strong bargaining position.