The Daily Mirror is reporting that both of the London clubs will try and sign the midfielder from the Toffees in the winter transfer window.
Barkley was heavily linked with a move away from Everton over the summer but a switch failed to materialise for the midfielder.
The same report claims that Chelsea made a £35m offer to sign Barkley in the summer but the England international, currently sidelined with a hamstring injury, ended up staying at Goodison Park.
The story, however, also says that Tottenham could have the “edge” over Chelsea in the race for Barkley’s signature because of Mauricio Pochettino’s track record of nurturing youngsters into the first team.
Chelsea ended up signing six players in the summer transfer window after wrapping up deals for Willy Caballero, Antonio Rudiger, Alvaro Morata, Tiemoue Bakayoko, Davide Zappacosta and Danny Drinkwater.
However, the Blues have made something of a mixed start to the new season and now find themselves in fourth place in the table, six points behind the leaders Manchester City after seven games.
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