The 17-year-old has broken into the first-team set up at Barca this season while also impressing for for Spain at national level.
His current contract is set to expire in 2023 and El Nacional have reported Chelsea are interested in securing his signature.
Owner Roman Abramovich is said to be a big fan of the youngster and sees him as a potential star of the future.
The Blues currently have Jorginho, N'Golo Kante and Mateo Kovacic as their starting options in midfield. with Saul, Ross Barkley and Ruben Loftus-Cheek as backup.
But manager Thomas Tuchel would likely see the appeal of adding Gavi to his squad, given his experience at such a young age.
Barcelona are keen to see Gavi commit his future to the club and his release clause will increase massively should he re-sign.
It's believed it'll be over €900 million - similar to that of fellow teenagers Pedri and Ansu Fati.
Watching Gavi in action, it's easy to see why the Blaugrana hierarchy and new coach Xavi are desperate to keep him around.
He started in Spain's 1-0 win over Sweden on Sunday as Luis Enrique's side booked their place at the Qatar 2022 World Cup.
A goal from Alvaro Morata secured automatic qualification for Spain but it was Gavi in the middle of the park who stood out.
Enrique paid tribute to his performance at full-time after initially receiving criticism for picking him.
Left on the bench were experienced options in Koke, Rodri and Mikel Merino.
Gavi,- ً (@LSComps) November 17, 2021
The new wonder of Spain and Barcelona!pic.twitter.com/h8o5u8WL0a
"If I decide to bring Gavi is because I believe I know him and because he can do good things," Enrique told reports per Football-Espana.
"And he has played a spectacular game in front of 50,000 fans."
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