Leeds were willing to pay £250,000 for Tate's services but it appears a loan deal has been struck between the two sides.
Swansea boss Michael Laudrup expects Tate to stay until the summer after confirming it in his pre-match press conference for Swansea'a game against Fulham.
Since his arrival at Leeds, Tate has started seven Championship games, with Leeds winning five. Despite stabilising the Leeds defence with his ability to pick a pass, the Whites have only managed two clean sheets.
Tate is due to return to the Liberty Stadium on New Year's Day after Leeds' game against Bolton.