Gilmour is seen as a star of the future, but with intense competition in midfield at Chelsea, it has been decided he will not be able to get enough top flight action to advance his development.
With N’Golo Kante, Jorginho and Mateo Kovacic in his way, he would be limited to cup games if he remained as part of Tuchel’s squad.
He is expected to be given a key role at newly-promoted Norwich as Daniel Farke’s side aim to avoid relegation after bouncing straight back up to the Premier League last season.
Gilmour will sign a season-long deal at Carrow Road.
He is one of a host of departures planned at Chelsea this summer as Tuchel aims to re-shape his squad.
Ross Barkley and Ruben Loftus-Cheek are set to depart, either on loan or in permanent deals, while Tammy Abraham should command the biggest transfer fee.
Olivier Giroud is a target for AC Milan – and should head to the Italian giants if a fee can be agreed, while Marcos Alonso and Emerson could also go.