The 30-year-old joined up with his new team-mates this morning to train for the first time in readiness for Saturday’s Elland Road clash with Crystal Palace and he is expected to go straight into the team.
He joins the club having made a handful of Premier League appearances this season.
Tate was a trainee at Manchester United and spent a short spell on loan at Royal Antwerp, but he went on to have two loan spells with Swansea before joining the club permanently in February 2004.
He cut his footballing teeth in Belgium, making 17 appearances, but after failing to figure at Old Trafford he headed to South Wales where he has clocked up 325 appearances for the Swans.
United manager Neil Warnock told the club website: “He is a leader and I felt these last few weeks we’ve lacked leaders. It’s great to have him in.
“We’re short on bodies in there, Jason is suspended, so he’ll play on Saturday.
“He’s got a great attitude, it’s a super club for him.”
Another signing is expected before this evening’s loan transfer deadline with West Brom winger Jerome Thomas being heavily linked with a move to Elland Road.