According to a report by La Informacion, the Real Madrid boss has already spoken to the Poland international's agent, Maik Barthel, urging him to push his client towards Stamford Bridge.
Lewandowski has emerged as one of the most in-demand forward's in Europe at the moment and is also on the radar of Dortmund's Bundesliga rivals Bayern Munich, who have already agreed a deal to sign Mario Gotze this summer.
Red Devils boss Sir Alex Ferguson has been credited with an interest in the 24-year-old, who could arrive at Old Trafford in a £25million deal if all goes to plan, with Manchester United having trailed Lewandowski for the past 18 months.
However, they could now face additional competition for his signature from Chelsea, if Mourinho leaves the Santiago Bernabeu to link-up once again with Roman Abramovich in west London.
Lewandowski's four goals which helped knock Real Madrid out of the Champions League this year have only added to his burgeoning reputation, with German newspaper Bild reporting that Mourinho actually texted the player directly after the 4-0 first leg defeat.
And, since the Portuguese tactician's contact with Lewandowski's representative, Barthel has spoken to Bayern vice-president Karl-Heinz Rummenigge to make the club aware of Chelsea's transfer interest.
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