Espanyol and former Watford boss Quique Sanchez Flores is Stoke's first choice to succeed Mark Hughes, multiple outlets reported on Thursday.
Reports say a Stoke delegation met Flores in Barcelona on Wednesday, and that Republic of Ireland manager Martin O'Neill is also under consideration.
Stoke sacked boss Hughes last weekend after the 2-1 FA Cup loss at League Two Coventry, which followed a run of seven losses in 10 Premier League games.
They were interested in Derby manager Gary Rowett until he signed a new contract with the Rams on Tuesday.
Flores' one season in English football to date saw him guide Watford to a 13th-placed Premier League finish in 2015-16, with the Hornets also reaching the FA Cup semifinals.
The 52-year-old then took charge at Espanyol in the summer of 2016, on a three-year deal, and they finished eighth in La Liga last term. They were 14th ahead of Thursday's match at Levante.
Speaking after leading Espanyol into the Copa del Rey quarterfinals on Thursday, Flores preferred to focus on his side's cup exploits rather than discuss himself.
Asked if he had received an offer from Stoke, Flores told a postmatch news conference: "I'm not going to talk about me on such an important day for Espanyol.
"It's a day for Espanyol, for the club, for the players, it's a day to celebrate for having qualified for the quarters. I don't want to talk about me, but to enjoy these moments. I want to continue being happy with my profession."
The ex-Atletico Madrid and Benfica boss added: "I'm very grateful to Espanyol, I've been frustrated sometimes because we haven't advanced as fast as desired, but that's because Espanyol haven't been able to, not because they haven't wanted to.
"Right now we're not thinking about anything other than today's game and Sunday's match against Athletic [Bilbao]."
O'Neill and his coaching team agreed contract extensions with the Football Association of Ireland last October.
But the 65-year-old said after the Republic lost their World Cup qualifying playoff the following month that he would need to have "a real think'' about his future, and still nothing has been signed.
Since the departure of Hughes and his assistant Mark Bowen, first-team coach Eddie Niedzwiecki has been in temporary charge at Stoke, taking training ahead of Monday's league clash with Manchester United at Old Trafford.
Niedzwiecki has been assisted by Under-18s boss Kevin Russell, who faced the media at a news conference on Thursday afternoon.
Russell said: "It's been a bit difficult with a manager gone who is really well-respected in the club.
"He was very liked, and it's sad when someone loses their job. It was a matter of picking them up on Monday a little bit -- there are players who were hurting, and rightly so.
"But as the week has gone on, we all know football -- it's just a matter now of getting on with it and trying to prepare for the new manager coming in, making sure the players are in the right frame of mind, or the game on Monday, whichever it is [that comes first].
"It was difficult at first but they have picked up over the last couple of days.
"Whoever comes in is coming into a fantastic football club. They will be a lucky manager.
"Eddie's in charge and I'm just helping him out. I don't think there is a timescale on it. Obviously the owners want to make sure it's the right person, which can be difficult. I'm sure they'll try to get it done as soon as possible.
"A new manager could come in tomorrow, and it's just a matter of us trying to make sure the lads are prepared, organised and ready to go.''
Defenders Erik Pieters and Bruno Martins Indi, sidelined of late by hamstring and groin injuries respectively, could be available for selection for Monday having returned to training this week.