The Canaries are set to complete the signing of the 19-year-old for £3.25million.
The transfer fee will be considerably the most Motherwell have received, with the previous record sale being the £1.75million Celtic paid for Phil O'Donnell.
The Parkhead side had an offer of £2.8million, potentially rising to £3.2million, accepted by the Steelmen last week.
However, Turnbull and his representatives rejected the contract offer on Friday prompting Celtic to issue the following statement on Twitter:
"#CelticFC was pleased to agree a record-breaking fee with Motherwell FC for David Turnbull and we have quickly followed this up with a fantastic offer to the player through his agent.
"Currently, we have been unable to come to an agreement with the player’s agent. #CelticFC is content that it has made a magnificent offer and it is now up to the agent and player to decide if the player wants to join Celtic. If not, we move on."
It was reported that the deal was back on at the start of the week, only for the Scottish champions to pull the plug on their offer, allowing Norwich to sweep in for the midfielder who netted 15 times in the campaign just past.
The contract offer made to Turnbull has been reported as a 20 times what he was on at Motherwell.
According to the Scottish Sun, Celtic were willing to pay the player £12,000-a-week. A mammoth increase on the £600-a-week he earned at Fir Park.
It is believed that the player and his agent wanted closer to £20,000-a-week.
Unai Emery has identified the Celtic ace as the solution to the club's issues at left-back.