Glover currently holds a similar role with the Hammers, but Leeds have moved to offer him a position at Elland Road.
Leeds owner Massimo Cellino had offered the role to Norwich City scout Steve Head last month, before deciding to withdraw the offer after finding out Head was due to go away on holiday.
Cellino is keen to get a new recruitment man on board as soon as possible to help manager Uwe Rosler revamp the Whites squad ahead of next season.
Glover faces an uncertain future at West Ham following the departure of Sam Allardyce as manager and with Tony Henry taking on the role of director of recruitment at Upton Park.
Glover is highly-regarded in footballing circles having also worked with Allardyce at Blackburn Rovers and his appointment at Leeds is expected to be rubber stamped in the next few days.