Rovers' prospective new owner Ahsan Ali Syed has vowed to back Allardyce in the transfer market should a takeover go through, and he has made no secret of his affection for the former England captain.
But, although Allardyce feels the 35-year-old is still capable of playing in the Premier League, he has reservations over Beckham's willingness to return to an English club after leaving Manchester United in 2003.
"Mr Syed says that he would like to spend it on youth, and then that he would like to sign David Beckham," he told Sky Sports News. "Obviously that's not actually youth, but that's like more of a PR signing that would bring a lot of attention to the club.
"To me it is about whether David is still good enough to play in the Premier League at his age, and he probably is. And the other factor is does he want to come to Blackburn and does he want to come back to England?
"In my experience the answer to that is no. He has never really wanted to play for anyone else other than Manchester United, that was the last time I spoke to him which was a number of years ago, and if he can't do that then he would sooner player somewhere else in the world of football."