The struggling Whites are expected to sack Javi Gracia on Tuesday, with former England boss Allardyce primed to take over for the final four Premier League games of the season.
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Gracia was only appointed in February and Allardyce would be Leeds' fourth head coach of the season after Jesse Marsch and temporary appointment Michael Skubala.
Sporting director Victor Orta, who hired Gracia, is also fighting to save his job after extensive boardroom discussions on Monday.
Tony Pulis, 65, and former United star Lee Bowyer, have also been considered as a caretaker options for Leeds.
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Former Stoke manager Pulis' last role was as Sheffield Wednesday boss, which only lasted a month in 2020.
Bowyer, 46, who spent seven years at Elland Road as a player between 1996 and 2003, was last in charge of Birmingham City where he was sacked in 2022.
Leeds have endured a woeful run of form and are sitting 17th in the table - level on points with 18th-place Nottingham Forest.
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They conceded 23 goals in April, a new Premier League record for goals against in a month, having slumped to 5-1, 6-1 and 4-1 defeats to Crystal Palace, Liverpool and Bournemouth respectively.
On Sunday night fans' group the Leeds United Supporters Advisory Board issued a vote of no confidence in both the club's board and manager Gracia.
Leeds next play on Saturday, against leaders Manchester City, by which time Big Sam could be in the dugout.
LEEDS FINAL FOUR GAMES
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Man City (A)
Newcastle (H)
West Ham (A)
Tottenham (H)