Bale is understood to have grown increasingly frustrated with his lack of game time at the Bernabeu this season, with the Wales international making just 21 starts across all competitions throughout the course of the campaign.
The 28-year-old's omission from the first leg of Madrid's Champions League semi-final with Bayern Munich raised more questions over his future, with Bayern themselves thought to be amongst the clubs monitoring his situation.
However, talkSPORT reports that Bale is being backed to remain at the club by Perez, who is unwilling to lose the player who joined for a then world-record fee in 2013.
Bale himself is also thought to want to stay in the Spanish capital and fight for his place under Zinedine Zidane, despite falling down the pecking order following an injury-plagued spell.
The Welshman has scored 82 goals in 184 appearances for Madrid since joining in 2013, helping the club to 10 trophies including three Champions League titles.