The Polish striker has only a year left on his contract and reportedly wants a fresh challenge away from the Bundesliga.
However, according to Sky in Germany, Lewandowski is not prepared to wait until he is a free agent to depart Bayern and is eyeing a move to the Nou Camp this summer.
Since arriving at the German champions in 2014, Lewandowski has scored 343 goals in all competitions.
In his time at the Allianz Arena, he has lifted eight Bundesliga titles, three German cups and the Champions League in 2020.
His departure would be difficult for Bayern to cope with especially if Serge Gnabry, another player with his contract running out next year, also leaves this summer.
Barcelona’s interest in Lewandowski is curious considering the Spanish giants snapped up Pierre-Emerick Aubameyang in January from Arsenal.
The Gabonese international has started well at the club, scoring 13 goals in 21 appearances so far.
If Barca’s manager Xavi was to sign Lewandowski he could shift Aubameyang to the left-wing, a role he often played at Arsenal.
Speaking earlier this year, Bayern chief executive Oliver Khan noted his club will be doing their best to keep Lewandowski beyond the summer.
He told Amazon Prime: ‘We would be stupid to sell a player who scores 30 to 40 goals per season. We are and will be in talks about his future in the next weeks.
‘He has a contract until June 2023 and we’ll definitely have him with us for another season.’
For more stories like this, check our sport page.