Gilmour put in several excellent performances for Scotland during the European Championships last summer. Expectations were high for the 20-year-old midfielder when he joined newly-promoted Norwich City on a loan deal for the 2021-22 season.
However, the campaign turned out to be a disaster, as the Canaries finished bottom of the table after winning just five top-flight games throughout the year.
Often used as a defensive midfielder, Gilmour didn't personally enjoy a great time as he played 28 times across competitions without scoring a single goal.
The young Scot is unlikely to play much for Chelsea next season with the likes of N'Golo Kante and Jorginho ahead of him in the pecking order. This means he may have to find another temporary move to aid his development further.
According to talkSPORT (as per The Scotsman), Chelsea icon Lampard is keen to pickup the starlet on a temporary basis to bolster midfield options for the Toffees.
Lampard, who kept the Merseyside club in the top flight after succeeding Rafael Benitez in January, is a big fan of Gilmour. He also wants to bring in fellow Blues midfielder Conor Gallagher, who spent the previous season on loan at Crystal Palace.
Gilmour has also been linked with a move back to his boyhood club Rangers and has just one year left on his current deal at Stamford Bridge.
Conor Gallagher gives view on Chelsea future following exceptional season at Crystal Palace
Conor Gallagher has had two incredibly successful seasons on-loan at Premier League sides West Bromwich Albion and Crystal Palace. The 22-year-old midfielder will be desperate to make an impression at Stamford Bridge, where he is yet to make a senior appearance.
While on England duty, Gallagher refused to rule out the possibility of going out on loan once again, as he told BBC's Football Daily (as quoted by The Mirror):
"I've not really had time (to think), to be honest. It's been a long season at Palace. I've only just finished really, then I've come here with England. That's all I'm thinking about for the next four matches here, and I want to do well. I've not really thought about it."
"I feel like if you look at my loans, I've done four loans now, and they've always progressed, and it's always been a new challenge. I always believe in myself, and I try to do my best on the pitch. I think confidence and belief is a part of that. I've got three older brothers who all played football."
Gallagher had a breakthrough season on loan at Crystal Palace this season. He scored eight goals and provided three assists in 34 league appearances.
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