Unai Emery's side, reigning Europa League champions and Champions League participants this term, have a recent track record in the loan market, having previously brought in Marcus Rashford and Marco Asensio.
Jose Mourinho has left the decision over Endrick's future in the 20-year-old's own hands, but the player already understands where he stands in the pecking order at the Bernabeu, says AS.
The Real Madrid head coach has stated his preference for a smaller squad of around 20 outfield players plus those currently injured, a stance seen in Spain as a signal to fringe players including Eduardo Camavinga and Raúl Asencio that they may need to leave or find loan moves this summer.
Endrick has been left as Madrid's third-choice central striker behind Kylian Mbappe and new signing Carlos Espi, with the club also short of options on the right flank following the arrival of Yan Diomande.
Mourinho sees Endrick more as a central striker than the wide role he played on loan at Lyon last season, but does not view him as the purely positional number nine he wants as backup to Mbappe.
The Brazilian is weighing his options having already had a positive loan spell at Lyon, scoring eight goals and providing seven assists in 21 appearances to help force his way into Brazil's World Cup squad.
Any move away from Madrid this season would need to guarantee regular first-team football and Champions League involvement, conditions both AS Roma and Aston Villa are understood to meet.
A loan would also suit Real Madrid, who are protecting the value of a signing that has cost close to €80m in total, including the original €35m fee plus €25m in variables paid to Palmeiras, on top of wages, bonuses and commission. The club are said to be considering covering part of Endrick's net salary of more than €5m a season to ease a move, as they did for Franco Mastantuono's loan to Fiorentina. Villa's project under Emery is seen by the striker's camp as among the more attractive options if Madrid sanction a temporary exit.