According to Sam Wallace of the Daily Telegraph, while there is uncertainty at Stamford Bridge over the future of manager Antonio Conte, the Blues are ready to push on with their recruitment nonetheless. Lewandowski is said to be top of the club's list.
It's reported the Poland international would likely cost around £100 million, and his arrival would potentially spell the end for Alvaro Morata with the London club.
"Although Lewandowski's age is not ideal for the kind of hefty price Chelsea would have to pay—he turns 30 in August—he does fit the bill for a proven European goalscorer and his arrival would also be a crucial way of persuading Eden Hazard, and Thibaut Courtois that Chelsea are serious about challenging again for the Premier League and the Champions League," Wallace wrote.
The Blues won the FA Cup on Saturday, although overall their season was one of disappointment, as they finished in fifth in the Premier League and were eliminated from the Champions League in the first knockout round.
Conte has admitted that not getting into Europe's elite club competition is far from ideal for the Blues, per Liam Twomey of ESPN FC:
Getting a player like Lewandowski at the end of an underwhelming season would be a way for Chelsea to make a statement. At Bayern, the 29-year-old has established himself as one of the best strikers on the planet.
Lewandowski's power, composure and anticipation in advanced areas make him difficult to stop.
As relayed by BT Sport Football, he's been productive in front of goal for the last three seasons:
For Bayern, replacing someone as effective would be a huge challenge. However, given his age, a figure in the region of £100 million would be tempting to accept.
The player himself may potentially be stimulated by a change of scenery too. After all, he's won the Bundesliga six times now—twice with Dortmund and four times with Bayern—and his style of play would be ideal for the Premier League.
Chelsea author Mark Worrall expressed some concern about a possible deal but believes the Blues should go for it:
Wallace also noted there is interest in Lewandowski from Real Madrid but doubted whether the Spanish giants would be able to fund a deal for the player. At this juncture, they would surely represent a more enticing prospect than Chelsea.
The Blues have the finances and the pedigree to tempt the Pole to the Premier League, but at this stage of Lewandowski's career, where he's in his peak years, it would be a big surprise if he was ready to join a team that wasn't competing in the Champions League next season.