United have approached Glover after pulling out of a deal to appoint Norwich City scout Steve Head as their new head of recruitment earlier this week.
Head appeared to have secured the job at Leeds after visiting Thorp Arch and reaching an agreement with owner Massimo Cellino but Cellino withdrew his offer after learning that Head was due to depart on an immediate holiday.
Leeds are pushing for signings after naming Uwe Rosler as head coach and Cellino said he had “no time” to wait for Head to start work at Elland Road.
The Italian has also spoken with Celtic scout David Moss in the past month but Glover - West Ham’s head of recruitment since 2011 - is emerging as Cellino’s preferred alternative.
Before moving to Upton Park, Glover worked as head of recruitment at Blackburn Rovers.
He is under contract at West Ham but the Premier League outfit are set for a summer of change after parting company with manager Sam Allardyce.
Leeds are revising their entire backroom team and news of their move for Glover comes as former Leicester City boss Rob Kelly prepares to become Rosler’s assistant.