The 28-year-old moved to the Emirates from Deportivo last August for £17m but struggled to make an impact last season, starting just twice in the Premier League.
The Spaniard’s game-time will likely be reduced even further after the Gunners signed Alexandre Lacazette from Lyon for £52m.
Perez was expected to leave this summer even before the arrival of Lacazette, who also took the number nine jersey away from him.
Now Perez's agent, Rodrigo Fernandez Lovelle, has revealed that Deportivo have made an offer to re-sign him less than 12 months after his departure.
Lovelle said: “I’m here to see the Emirates Cup and to be with Lucas.
"We have talked to everyone. Arsenal know we want to go, that there are no opportunities here, and within a year there is a World Cup.
"Would you not be angry and sad and upset if you had your squad number taken without being asked?"
Lovelle added: “Deportivo made an offer a week ago. It is a permanent deal and worth more than Nolito's move to Sevilla.
"Of course [he was told he would play more]. Lucas signed for Arsenal and was the happiest person in the world and with all of his dreams to fulfil.
"We have seen that every time he played he did well, but after every time he got fewer and fewer opportunities."
Arsene Wenger spoke of Perez's situation towards the end of last season while his own future at the club was up in the air.
The Frenchman has since signed a new two-year deal but looks set to allow Perez to leave.
Wenger said in May: “If I am the manager I would love to keep him but we have to sit down and see if he has a realistic chance to play.
"He's a top quality striker but I couldn't give him the games he wants and deserves, I confess that. I feel a bit sorry for him."
