The Blues made two big attacking additions during the last transfer window in Raheem Sterling and Pierre-Emerick Aubameyang, but Nkunku was also linked with a move to Stamford Bridge.
And the two-time European champions appear to have stepped up their pursuit of the France international, who is said to have undergone a private medical in Germany.
According to transfer guru Fabrizio Romano – writing in his Daily Briefing for Caught Offside on Sunday – Chelsea have tabled an offer which meets Nkunku's satisfaction.
Nkunku scored 35 goals in all competitions last season as Leipzig finished fourth in the Bundesliga and won the DFB-Pokal (German cup) – where he equalised in the final – for the very first time.
He ended a five-match goal drought on Saturday as Leipzig thumped Bochum 4-0 (Timo Werner, who left Chelsea to rejoin the Red Bulls last month, got the other two goals).
During the last international break before the World Cup, Nkunku won his seventh and eighth caps for France, coming off the bench as Les Bleus beat Austria then lost to Denmark in the Nations League.
Despite showing great goalscoring prowess at club level, however, he's yet to find the net for his country.
More Chelsea stories
Conor Gallagher's 90th-minute winner at Crystal Palace on Saturday secured Graham Potter's first win as Chelsea boss – but Thiago Silva was fortunate not to see red for a handball.
Potter has reportedly given his approval for Chelsea to sign Declan Rice, who has been linked with a move to Stamford Bridge for some time. Inter Milan centre-back Milan Skriniar is also said to be a target for the Blues.
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