Staunton, like McAllister, joins the club initially until the end of the season and completed his deal at Elland Road after overseeing training earlier on Monday.
"I'm delighted to be here," Staunton told the club's official website. "Gary brought me to Coventry as a player, I played with him at Liverpool, and it's nice to team up again.
"I hope I can be of assistance and help this great club get back to where it belongs. It's a great opportunity to work at a big club with a good manager, and learn the ropes from someone else on the inside and see how a club runs.
"Your next progression after playing is coaching and managing and this is a great opportunity for ourselves to be part of something that will be big in a few years."