According to RMC Sport Journalist Fabrice Hawkins, the 24-year-old has completed his medical but nothing has been signed between the two parties yet.
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The Frenchman has been in scintillating form for the Bundesliga side following his £11.7million move from Paris Saint-Germain in 2019.
Nkunku, 24, has scored 53 goals in 147 games for Leipzig and been linked with a move back to the French champions as well as Juventus and Liverpool and Manchester City.
But the Blues look to have taken the lead in negotiations and despite nothing concrete having been signed, Chelsea are 'confident' of tying up the £53m (€60m) deal.
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Hawkins insists Liverpool are still in the race for Nkunku but it remains to be seen whether they will step up their interest at this time.
Nkunku's form since arriving in the Bundesliga has seen him break into Didier Deschamps' France Squad and will be in the mix for a place in his World Cup squad.
He has eight caps for Les Bleus after making his full international debut against the Ivory Coast in March.
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This season, Nkunku has scored four times in seven games with Leipzig sitting in 12th in the table.
The forward started his career at hometown club PSG but was unable to claim a place in the first-team after the signings of Neymar and Kylian Mbappe in 2017.
Despite showing promise when he was given the opportunity, Nkunku was deemed surplus to requirements after the club's 2019 Coupe de France final, when he fired his spot-kick over the bar as Paris Saint-Germain were beaten on penalties by Rennes.
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