Manchester United rejected an opening offer of £27.6million plus James Rodriguez for the midfielder last week.
But reports have suggested that United are fearful of losing the World Cup winner with Real Madrid having until September 2 to try and snatch the star from Old Trafford.
The France international has hinted at a move for nearly 18 months, having called Los Blancos “a dream club for every player” back in March 2018.
United’s club-record signing, who cost them £89m in joining from Juventus three years ago, also said this June: “It could be a good time to have a new challenge somewhere else.”
Last month, Paul Pogba’s agent Mino Raiola publicly declared that the club are aware of his client’s wish to leave.
Real Madrid have long been tipped as a destination for Pogba, having been in the mix of clubs to sign him before he left Juventus to rejoin United in 2016.
And Spanish daily newspaper Marca claim that the La Liga giants could look to include up to three players in an offer as they look to tempt United into business.
Ole Gunnar Solskjaer is determined to keep hold of the Frenchman but ZInedine Zidane is desperate to link up with his countryman at the Bernabeu.
After Rodriguez, Madrid could look to include out-of-favour winger Gareth Bale in a deal, say Marca.
The Welshman almost moved to China last month after Zidane made it clear he doesn’t want to keep the 30-year-old.
But the deal fell apart with Chinese Super League outfit Jiangsu Suning unwilling to pay a fee to Los Blancos for the player, who they were set to hand a contract worth £1m-a-week.
Isco is another player whose future has often been uncertain, though Manchester City boss Pep Guardiola is also an admirer of the Spaniard understands Express Sport.
Mariano Diaz is another player the Madrid hierarchy want to offload though whether he would appeal or not to United is another matter.
United are however planning a potential swoop for Tottenham creator Christian Eriksen due to their fear Madrid will make another approach for Pogba.
The Dane has long been on United’s radar and the 27-year-old is out of contract in a year’s time, meaning he could be available for well below Spurs’ normal valuation.
Spurs chief Daniel Levy had quoted teams £130m for Eriksen at the start of the window but he could be forced to accept half that if he wants to avoid losing the player for nothing next summer.