The London club have allegedly been in contact with the former Manchester United and Real Madrid hero, who will be available on a free when his contract expires in November
The Daily Mail is claiming there has been initial contact been Beckham’s representatives and the newly promoted club.
Fernandes tweeted: “I think having someone like David Beckham in the club would be great. Was always in my thoughts. Let’s see.”
The 36-year old’s MLS contract expires in November, which means he can leave LA Galaxy on a free transfer.
Meanwhile, Bookmakers William Hill have slashed their odds on the veteran midfielder returning to England to 4/7, with QPR priced at 5/6 to snap him up this season.
"It may well be that David Beckham would like one last hurrah in the Premier League before he retires, and reports suggest that he could do so for QPR, for whom it would be something of a coup," said the company's spokesman Graham Sharpe.
There is believed to be a whole host of clubs interested in bringing him back to European football with Tottenham, Arsenal and AC Milan all reportedly monitoring the situation.
QPR boss Neil Warnock has not been shy in splashing the cash since Fernandes took over last month, with Joey Barton, Shaun Wright-Phillips and Anton Ferdinand all moving to Loftus Road late on in the transfer window.
Signing Beckham would be a great marketing opportunity and Fernandes is already reportedly eyeing up establishing a QPR brand in Asia, where the former Manchester United star is hugely popular.
However, according to some reports, Beckham will not think about his future until his contract expires.