But Madrid are yet to find their best form in pre-season, despite signing the likes of Eden Hazard, Luka Jovic, Eder Militao and Ferland Mendy.
If the accepted fee is in that expected region of €60m (£55m), then it would make the 22-year-old nearly £100m cheaper than Pogba.
"From what I've learned, the thing goes further: there is already a proposal accepted by the Dutch club and player", Bechler tweeted on Friday morning.
But Van de Beek refused to sign a new deal and is now waiting to see whether Real Madrid make a move for him in the coming weeks. "Offsets the few goals of Madrid strikers".
The Manchester United midfielder has been linked with a move to the Spanish capital for a fee reported to be around £150 million.
By rejecting the new deal, Van de Beek has made it clear that he is prepared to jump if a big club make an offer for him.
Their team has been ripped apart since the last season, with Matthijs De Ligt and Frenkie de Jong among the stars to depart the Johan Cruijff ArenA this summer.
Ajax begin the defence of their Eredivisie title against Vitesse on Saturday and in the build-up, Van de Beek was asked about the Real Madrid rumours.