The Spanish giants are selling Casemiro to Manchester United on a deal which could net them £70m.
Carlo Ancelotti has insisted he can cope with Casemiro’s departure, but Real Madrid are still checking what they could do to replace the Brazilian.
They have been linked with holding an interest in Spurs midfielder Hojbjerg and are now considering going in with a bid.
The offer Real Madrid are considering making is claimed to be £45m.
Tottenham boss Antonio Conte is a firm admirer of Hojbjerg, but has a host of options in midfield and could be tempted to cash in on the Dane, who turns 28 next month.
Spurs snapped Hojbjerg up from Southampton in the summer of 2020, paying an initial £15m to secure his services.
Selling the midfielder for £45m would represent a substantial profit and also give Conte extra funds to chase further new faces before the transfer window closes on 1st September.