Fir Park chiefs accepted an offer from the Premier League new boys on Tuesday night on the same terms previously offered by Celtic.
And it is understood the Scotland Under-21 playmaker, who scored 15 goals for his boyhood club last season, is now travelling south to open discussions with Canaries boss Daniel Farke over a move to Carrow Road.
The Hoops pulled out of the running to sign the 19-year-old midfield ace after issuing a take-it-or-leave ultimatum last week.
They said they had made Turnbull a “magnificent offer” but were ready to “move on” to other targets if Turnbull was not ready to accept.
Well bosses were hoping to cash in a “transformational” sum that would secure the fan-owned club’s future, but that was threatened when Celtic finally backed out on Tuesday night.
However, Norwich have stepped forward to offer an up-front bid of £3m, with further payments to come if additional clauses are triggered – including a sell-on fee.
If Turnbull does agree to join Farke’s side, the Scottish Football Writer’s Association’s Young Player of the Year could find himself lining up alongside Scotland internationals Kenny McLean and Grant Hanley as well as former Celtic forwards Patrick Roberts and Teemu Pukki.
Aberdeen have confirmed the signing of Derby midfielder Craig Bryson.
The 32-year-old has spent the last eight years at Pride Park, making almost 300 appearances.