The Whites are looking for a new shooter after allowing Kiko Casilla to join Spain’s Elche team on loan.
Casilla was the club’s number one for much of its promotional campaign, but was banned in February last year for abusing Charlton’s Jonathan Leko.
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Leeds signed Illan Meslier from Lorient five months later, the 21-year-old asserting himself as Marcelo Bielsa’s first-choice goalkeeper.
Klaesson was shortlisted by club football manager Victor Orta for some time with a reserve keeper being a top priority this summer.
Newcastle United’s Freddie Woodman and Levante star Dani Cardenas have also been linked with a move to Elland Road.
However, according to Football initiate, Leeds did accept an offer for the 20-year-old.
In addition, the Yorkshire club have reportedly already applied for a work permit for Klaesson.
Elsewhere, former Tottenham midfielder Jamie O’Hara claimed that Bielsa is a “world class” manager who “needs” to move to a big club.
“He must be at a big Bielsa club, right?” He has to be at a club like Barcelona, let’s be honest, ”said O’Hara TalkSPORT.
“Spurs are a bigger club than Leeds… that would be a step forward. Jose Mourinho isn’t going to be Leeds manager, is he?
“I’m not saying Leeds are not a big club and I’m not saying Bielsa has to leave, I’m saying he should be at a club like Barcelona. He’s a great manager and Leeds are very lucky to have him.
“I would love him at Spurs, but he will never leave Leeds to go to Spurs. Not enough ambition among Spurs.
“He’s a world-class manager.
“I think he has some kind of love affair with this club, the city and the fans. They all love him and I think he loves them.
“But if someone like Barcelona, Real Madrid or PSG comes knocking, I’m sorry you don’t stay at Leeds United, you go. “