The Scotland star has agreed a loan move to the Premier League new boys from Chelsea.
According to Sky Sports, he is to undergo a medical at Norwich ahead of the transfer.
Gilmour was man of the match for Scotland in the 0-0 draw with England at Wembley in Euro 2020 before testing positive for Covid-19 prior to the third and final group stage match against Croatia.
The 20-year-old was a key miss in the midfield for Steve Clarke as he self-isolated.
Gilmour has been linked with a return to Rangers on loan.
The player left Ibrox as a teenager for Chelsea and has since played 22 times for the first-team.
Manager Thomas Tuchel is keen for the player to get first-team football and is currently behind Matoe Kovacic and Jorginho in the pecking order at Stamford Bridge.
At Norwich, Gilmour will link up with Scottish internationals Kenny McLean and Grant Hanley.