Quique Sanchez Flores has been given a Friday deadline to decide whether he wants to take over at Stoke, according to Sky Sports' Spanish football expert Guillem Balague.
Balague says Stoke have already met with the Espanyol head coach in Barcelona ahead of a potential appointment.
Flores has been at Espanyol since the summer of 2016 after he left his position as Watford head coach.
Balague believes a return to the Premier League could happen after Stoke offered the Spaniard a five-year deal at the bet365 stadium.
He said: "They [Stoke] want an answer by tomorrow. Quique Sanchez Flores has not said yes yet but will answer tomorrow. He will either give them an immediate yes or he will stay at Espanyol.
"My instinct is that the proposal is so interesting plus the fact that Espanyol have not kept their promises from when Quique first arrived suggests to me that he will announce his departure from Espanyol."
The 52-year-old proved a success during his previous spell in England. He reached the semi-finals of the FA Cup in 2016 and also guided Watford to 13th place in the Premier League.
Despite his success, he was replaced by Walter Mazzarri at the end of his solitary season in England.
Weeks later, he signed a three-year contract as Espanyol head coach, he has 18 months left on the deal.