Thomas Tuchel's side were strongly linked with a move for Borussia Dortmund star Erling Haaland, but have had their efforts rebuffed repeatedly.
According to reports, the Blues are now considering making an offer for Lewandowski, who has two years left on his deal at the Allianz Arena, but has been linked with an exit this summer.
Lewandowski has also been linked with a move to Premier League champions, Manchester City, who are looking to replace Sergio Aguero, who joined Barcelona on a free last month. Pep Guardiola worked with the 32-year-old during his three years at Bayern Munich, winning two Bundesliga titles and the German Cup, and is reportedly keen to be reunited with his former player.
Chelsea, meanwhile, are looking to close the gap with City, as well as bring in a new striker, after Timo Werner failed to impress last term.
Real Madrid have also been long-term admirers of Lewandowski, with his agent Pini Zahavi recently brokering a deal that saw David Alaba move from Bayern to Madrid this summer. Lewandowski won the UEFA Men's Player of the Year award in 2020, and scored 48 goals in 40 games across competitions last season as Bayern won their ninth Bundesliga in a row.
New Bayern manager Julian Nagelsmann admitted he wasn't happy with having to deal with Lewandowski's future immediately after taking over from Hansi Flick. "The rumours about Robert Lewandowski have been around for ages, I think since he joined Bayern Munich," he said to Sky Germany.
"I think it's normal for someone who scores so many goals that almost every club puts out a feeler. I think on one hand Robert knows what he has in Munich, what he has in this team. I've spoken to him and texted him. I'm not very fond of the first official act with a player being to immediately discuss his contractual situation."