The Yorkshire giants have agreed a fee with Valerenga for the services of the highly-rated shot-stopper, who has flown to England; he arrived on Sunday.
He is, according to Norwegian outlet Nettavisen, quarantining in England and will be finished with his isolation period on Thursday.
Leeds are planning for Klaesson to undergo a medical check on Friday and if everything goes according to plan he will put pen to paper to a contract with the club.
The Whites identified the goalkeeper in 2019 at the same time as they signed Illan Meslier and believe he has great potential.
They are looking for Klaesson to provide backup to Meslier and also push him for the number 1 spot at Elland Road.
Leeds are to pay in the region of £1.6m to sign the goalkeeper and Valerenga have inserted a series of add-ons into the agreement.
The Premier League side have been looking for another goalkeeper since allowing Kiko Casilla to join La Liga club Elche on loan earlier in the transfer window.