Louis van Gaal says Manchester United have offered David de Gea "a lot of money" in a bid to keep him at Old Trafford.
Beckham's five-year stay with Major League Soccer franchise the Los Angeles Galaxy comes to an end this weekend and, at first glance, a move to the second tier of French football may seem an unlikely fit for the former England captain.
But since their relegation from the top flighr in 2010, Monaco have been taken over by billionaire Russian businessman Dmitry Rybolovlev, who has lofty ambitions to restore them as a major force.
Karlsen told BBC Sport: "We are happy with our current squad but if there's a chance to get a player of Beckham's stature, we'd be foolish not to explore it.
"Right now we're trying to learn if Beckham's future ambitions are in sync with ours.
"I understand he's high in demand but it's natural for a player with his experience and quality. Let's see what the next few weeks bring.
"It's a privilege to be linked with world-renowned players like Beckham."
Sources close to the player have confirmed an approach by Monaco, who are an ambitious club with wealthy owners, but that they are not regarded as favourites.
Beckham would probably prefer to go to Ligue 1 heavyweights Paris St Germain if a suitable offer is forthcoming and team up again with coach Carlo Ancelotti and team director Leonardo, whom he knows from his two previous loan spells with AC Milan.
There are also offers from China, Russia and Brazil on the table which Beckham will consider after the Galaxy's MLS Cup final against Houston.
Beckham told Sky Sports News on Thursday he is weighing up a number of offers, but does not anticipate a swansong in England.
"I still feel I can play and I'm not considering retiring," he said.
"I do believe I still have got another challenge inside, that I can still play for.
"I've still got managers like Carlo Ancelotti and Harry Redknapp and Sam Allardyce coming out that are saying they'd love me to play and they know how professional I am, it means a lot to me.
"It is what I have tried to base my career around, over the years, being professional and working hard, so it's nice to be recognised like that.
"I've always said I think I would struggle to come back and play in England because I've played for the biggest club in the world, the biggest club in England, Manchester United, and I couldn't see myself playing for any other Premiership team.
"But you never know. Like I said, we've got some exciting options on the table. I do think we'll be spending a little more time in England because it's good for the kids to see their grandparents."