The Scotland international, who was named man of the match on his full national debut against England last week before missing Tuesday's defeat to Croatia after a positive coronavirus test, has attracted interest from a number of Premier League clubs.
Southampton were keen on a deal in January before Chelsea ruled out an exit and have kept tabs on the 20-year-old midfielder in the months since, though it is thought the south coast club have not made any real approach this summer.
Chelsea are keen on sending Gilmour to newly promoted Norwich as their manager Daniel Farke has a good relationship with Tuchel after the pair worked together at Borussia Dortmund.
Sources close to the deal say no final decision has been made over the move, though it is understood the Canaries are the clear frontrunners.
Gilmour remains in isolation following his positive test for coronavirus this week, though it is not proving an issue when it comes to talks progressing.
Norwich are on the look out for a midfielder after Oliver Skipp left the club following his loan from Tottenham last season.
Gilmour made just five Premier League appearances last season having missed the start of the campaign through injury before being given limited chances by Tuchel, who is looking to trim a bolstered squad this summer.
Ruben Loftus-Cheek is another who could be sent out on loan after his underwhelming spell with Fulham last season.