Reports in Germany carried by ESPN claim that the Poland international must make a decision by May or face another season at Dortmund.
Lewandowski is under contract with the Bundesliga club until the end of the 2013-14 season, but he's refused to sign an extension and has been linked with a move to the Premier League leaders this summer.
Manchester United are one of several clubs thought to be interested in the 24-year-old, with the report claiming Bayern Munich, Chelsea and even Arsenal are monitoring developments.
But Dortmund say they will not entertain offers for their star striker, who leads the Bundesliga golden boot race with 20 goals, after May and the club's general manager Michael Zorc told Kicker he's not received any offers so far.
Zorc said: "We have clearly positioned ourselves. We hear there is interest. But until now we have not received any offers."
Reports have suggested Bayern Munich are in pole position to sign the Dortmund striker, with speculation claiming a deal for the forward is already agreed.
But there's been no confirmation from either party. The report also suggests Lewandowski will command a fee around the €30m mark, and with interest from the likes of United, Chelsea and Bayern Munich, it wouldn't be a surprise to see that figure driven higher.
Lewandowski has five goals from nine Champions League appearances this term, but failed to find the net during Wednesday night's 0-0 draw with Malaga at the quarter-final first-leg stage.
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