Besiktas president Fikret Orman has said he would not stand in Guti's way if Real Madrid wanted him to replace Julen Lopetegui.
Madrid go into Sunday's Clasico at Barcelona with Lopetegui under increasing pressure after taking only one point from the last 12 available.
Former Madrid midfielder Guti, who took over as assistant to head coach Senol Gunes at Turkish side Besiktas in July, is reported to be among the candidates to succeed Lopetegui if he is sacked.
"It would give us great pride if Guti returned to Madrid," Diario AS reported Orman as telling a news conference .
"He deserves that role -- we were keen for him to take the coaching post here the previous summer.
"No one has contacted us about Guti so far. We haven't heard anything about it but, if we receive a request from Real Madrid, we'll be happy about it."
Guti, 41, coached Madrid's youth teams after ending his playing career in 2011. He scored 77 goals in 542 games for the club between 1995 and 2010, and has spoken of his ambition to coach the senior team.
He had been considered a frontrunner to succeed Zinedine Zidane following his departure during the summer, but Madrid opted for Lopetegui instead.