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The 22-year-old has attracted plenty of attention, with Tottenham and West Ham interested, while Brentford saw a £35m bid for the forward rejected.
The Evening Standard report that the Blues fancy their chances of getting Johnson for £40m, although Forest have previously valued him at £45m-50m.
Johnson was a big part of the Forest team which won promotion from the Championship in 2022 and remained key for Steve Cooper last season, scoring eight goals in 38 league games.
He started the first game of the season last weekend and spurned a great early chance for Forest as they went on to lose to Arsenal at the Emirates.
Chelsea boss Mauricio Pochettino is keen to add another forward to his squad, especially after the significant knee injury suffered by new signing Christopher Nkunku who is likely to be out till December.
Another summer signing, Nicolas Jackson, started in the centre-forward role against Liverpool on Sunday, but Pochettino does not have a wealth of options behind the man signed from Villarreal.
The Blues were confident of bringing in Michael Olise from Crystal Palace, but the 21-year-old has opted to sign a new deal with the Eagles instead.
The Blues made a £35m bid for Olise which met his release clause, but the winger did not respond and has instead signed a new four-year contract at Selhurst Park.
MORE : Mauricio Pochettino wants Conor Gallagher stay but Chelsea stall new contract and haven’t offered fresh terms
MORE : West Ham push for £40m Mohammed Kudus deal as they fear Chelsea hijack
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