The Yorkshire giants are without a permanent manager following the sacking of Jesse Marsch and are looking for a replacement.
They have shortlisted a number of candidates and have already interviewed as many as four managers, though a solution is yet to be found.
Gallardo is yet another name that has been linked with a move to Elland Road, with the Whites coming up with a concrete approach, according to Argentine outlet TyC Sports.
Though Gallardo thanked Leeds for expressing their interest in him, he has insisted that he does not want to take over in the middle of the season.
Gallardo is without a job after leaving River Plate at the end of the year and is keen on taking a break before resuming.
The Whites will now have to look for a solution elsewhere with a number of names being linked.
Feyenoord coach Arne Slot has already rejected the post, while their attempts to land Andoni Iraola have hit a roadblock.
