The Blues have scored just two goals in their opening two Premier League fixtures and they look light up front following injury to Christopher Nkunku.
Nicolas Jackson is yet to open his account for the Blues and Mauricio Pochettino is keen on adding more firepower to his attack.
Johnson impressed during his debut Premier League campaign last season, scoring ten goals in 44 appearances in all competitions.
That form has attracted attention from around the Premier League, with Tottenham boss Ange Posecoglou said to be an admirer of the 22-year-old.
But Chelsea had made an approach to Forest last week, in a bid to beat their rivals to Johnson’s signature.
However, Chelsea have been put off by Forest’s asking price of £50m and will focus their attention elsewhere in the remaining week of the window.
That could open the door for rivals Tottenham, who are looking to replace Harry Kane following his £86m to Bayern Munich.
