The striker's representative Maik Barthel has confirmed that offers have been received from some of England’s top flight sides but insisted that, with interest continuously growing, clubs will need to up their bids in order to land the forward.
"Unofficially there are offers from English top clubs." Barthel told Goal.com. "We respect the contract situation in Dortmund and don't want to take part in speculation. A 13 million pound transfer fee is not suitable."
The 23-year-old has been strongly linked with a move to Manchester United this summer, where a deal would see him link up with former teammate Shinji Kagawa, who joined the Red Devils from Dortmund in June.
However, Sir Alex Ferguson has recently confirmed the club's interest in Arsenal's wantaway striker Robin van Persie, who revealed his desire to leave the club in an open statement to fans, possibly instead paving the way for the Gunners to bid for the Polish forward.
Reports on Saturday morning also suggested that last year's Champions League winner, Chelsea, has joined the race to secure Lewandowski's signature, but that it had an offer of around 13 million pounds turned down.
The Blues are thought to be looking to replace Didier Drogba after his departure to Chinese club Shanghai Shenua and turned to the Pole after Napoli's Edinson Cavani proved an unviable option.
Lewandowski scored 22 times in 34 games in the Bundesliga last season, adding seven goals from six appearances in the German Cup, including a hat trick in the final as Dortmund crushed Bayern Munich 5-2.
The player has been the subject of speculation since before Euro 2012, and went on to impress for co-host Poland during its short-lived time in the competition, netting in the opening game of the tournament against Greece and firmly putting his name in the shop window.