Nkunku had already agreed a pre-contract with the Premier League club in December and has now rubber-stamped his move to Stamford Bridge.
Eintracht Frankfurt came from a goal down to book a date with RB Leipzig in the German Cup final with a 3-2 win over Stuttgart on Wednesday.
City are also playing under a cloud after being charged with more than 100 breaches of financial rules by the Premier League
Manchester City midfielder Kevin de Bruyne will miss Wednesday's Champions League last 16 first leg with RB Leipzig, after he was left out of his side's squad for the game.
RB Leipzig striker Timo Werner said on Wednesday he did not have any regrets over his decision to move to Chelsea rather than German giants Bayern Munich.
Chelsea's Germany striker Timo Werner has signed a four-year contract to return to Bundesliga side RB Leipzig, the two clubs announced on Tuesday.
German champions Bayern Munich survived a late comeback from RB Leipzig to claim a 5-3 victory in the German Super Cup on Saturday with an impressive first half attacking display and new signing Sadio Mane scoring on his debut.
Yussuf Poulsen's late goal sealed ten-man RB Leipzig's 1-0 victory over Bayer Leverkusen on Saturday to trim Bayern Munich's runaway lead to seven points with seven Bundesliga games left.