Real join United in pursuit of Dembele
Arsenal and Tottenham have also been linked with bids for the 25-year-old, who shone for Fulham last season after moving from his usual role as a second striker to central midfield.
The Belgian has made a huge impression since his £5million move from Dutch side AZ Alkmaar in 2010 and could leave the club for nothing next summer.
The club are reportedly holding out for at least double that fee as Clint Dempsey is also tipped to leave and they will need to increase their attacking options ahead of the start of the Premier League.
But Fulham boss Martin Jol is facing a tough fight to keep the attacking midfielder at the club with the biggest sides in Europe chasing his signature, according to his agent Jan de Visser.
"Real Madrid have contacted us and asked about the possibility of signing Moussa Dembele," De Visser, of management agency Sport International, said in De Telegraaf.
"They are not the first big club who have contacted us this summer for him."