The Brazilian midfielder wants to make a decision before the end of the week.
Sources have told ESPN that a meeting took place on Wednesday evening between Casemiro's entourage and Real Madrid.
Casemiro, 30, will also speak to Madrid coach Carlo Ancelotti in the next few hours to learn more about the role he would have in the team following the arrival of Aurelien Tchouameni. The former Monaco midfielder started ahead of Casemiro in Madrid's opening LaLiga game of the season against Almeria on Sunday.
According to sources, Ancelotti wants Casemiro to remain at the Bernabeu but will accept whatever decision is taken by the club and the player.
Real Madrid, meanwhile, are waiting for United to make a formal offer for Casemiro.
Several sources have told ESPN that Madrid will not put up any obstacles if the player wishes to leave and United make an offer of at least €65 million.
Sources told ESPN that Casemiro was not ruling out the option of leaving Madrid.
The Brazil international had initially decided not to leave Madrid with the season having already begun. However, sources have told ESPN that Casemiro is gradually changing his mind because United offers him a much higher salary than he receives at Madrid.
Casemiro is under contract with Madrid until June 2025 and has won five Champions League titles.
The player knows that United's offer could be his last big contract, which is why his entourage has asked Madrid to listen to United's approach and expect the Spanish giants to accept Casemiro's decision.
During Madrid's league title celebrations in April, ESPN reported that Casemiro was considering ending his time at Madrid, having received interest from several clubs. During the summer tour in the United States, Ancelotti assured Casemiro that he was ahead of Tchouameni in the pecking order.
Real Madrid have been linked with a move for Newcastle United's Brazil international Bruno Guimaraes as a potential replacement for Casemiro, however sources have told ESPN Brasil that Newcastle have no intention of selling the midfielder.
Man United have lost their opening two Premier League games and are at the foot of the standings.
Information from ESPN Brasil's Gustavo Hofman was used in this report.