Marcelo Bielsa is reportedly aiming to sign the Bosnian midfielder on a permanent deal before the transfer window shuts at 5 pm on Thursday.
The report claims that Leeds are planning to submit a bid in the region of £4.5million for the 25-year-old defensive midfielder.
Besic joined Everton in 2014 but his career never really kicked on at the Merseyside club, with the player suffering a cruciate ligament injury on his right knee during the 2016-17 season.
Last season, he was loaned out to Middlesbrough where he managed 17 appearances in all competitions.
The Mirror adds that if Bielsa fails to sign him permanently today, he would take Besic on loan for the season before making the move permanent next summer.
Would be a superb signing for Leeds
Besic is a fantastic player who was once described by Roberto Martinez as a “very complete footballer”. He is technically gifted and all he needs is regular football under his belt.
Leeds have signed five players this summer – including Patrick Bamford and Barry Douglas on permanent deals – and adding Besic would represent a fantastic transfer window for the Yorkshire club.
The Whites have made an impressive start to the new campaign, with Leeds managing a 3-1 opening day win over Stoke City. But, Bielsa still feels that he needs a defensive midfielder, and Besic would fit the bill nicely.