According to reports in Spain, the La Liga giants will offer £30million in a bid to bring the Uruguayan international to the Bernabeu this month.
Mourinho is hoping to take advantage of the furore currently surrounding the Liverpool striker, after Suarez was banned for eight games following an alleged race row with Patrice Evra.
The South American sensation was also banned for a further game and fined £20,000 for a two-fingered salute towards Fulham fans at Craven Cottage.
Reds boss Kenny Dalglish has defended Suarez to the hilt, and the club has come into criticism for its staunch backing of the player.
"Obviously there was a big issue with Luis," Dalglish told the club's official website. "The players showed support for Luis which was fantastic, but then some people interpreted that wrongly as the players saying they're not interested in the fight against racism.
"That is totally and utterly rubbish. We don't want racism anywhere near football and certainly not anywhere near this football club."
After opting earlier this month not to appeal against the length of Suarez's punishment, Liverpool will be without the star striker until mid February.
Madrid are now hoping they can tempt the Anfield-based club into a sell with a lucrative offer before the close of the transfer window.