Wolves' Bakary Sako has revealed he is mulling over offers from the Premier League but is yet to decide on his future.
Rotherham boss Steve Evans will decide the fate of his out-of-contract players in a meeting on Monday.
Tate, 30, teamed up with his new team-mates on Thursday and is expected to go straight into the side for Saturday's clash against manager Neil Warnock's former club Crystal Palace at Elland Road with Jason Pearce still suspended.
"He's a lad who I've always enjoyed playing against," Warnock told the club's official website. "He's 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 and it's a super club for him."
Tate, who has made four appearances for Swansea in the top flight this season, began his career at Manchester United before sealing a permanent move to Swansea in February 2004 following two loan spells. He has made a total of 325 appearances for the Welsh club.