The 39-year-old is expected to receive a short-term contract until the end of the season, in line with the deal agreed with McAllister.
Staunton had a long playing career which included two spells with both Liverpool and Aston Villa, and he won 102 caps for the Republic of Ireland during an international career spanning 13 years.
He was taken on as the Republic's manager in 2006 but had his contract terminated by mutual consent in October after the Irish failed to qualify for Euro 2008.
Sammy Lee and Gary Speed had both been linked to the assistant's job at Leeds, but United will aim to tie up a deal with Staunton in time for Saturday's visit to Northampton Town.
Staunton played under McAllister briefly at Coventry City in 2003.
A statement from the club read: "Steve Staunton joined the squad on Monday with a view to assisting Gary McAllister until the end of the season."