The 25-year-old forward, who flew into England on Wednesday after agreeing an 18-month deal with the Wildcats, has revealed he had been poised to sign for the Tigers until the sacking of coach Ian Millward.
"It was pretty close, all I had to do was put pen to paper," he told BBC Radio Leeds.
"I got a phone call from them saying that they were looking for a new coach and they had to postpone the move.
"As soon as that happened, Wakefield called me up and asked me if I was interested in playing here.
"I spoke to a few of the players and the coach and I got a really good vibe from them."
Tautai, who played with Wakefield prop Justin Poore at Parramatta, has been named in coach Richard Agar's 19-man squad, along with fellow Samoan Ali Lauitiiti, Reece Lyne and Matty Wildie.
Agar has responded to his side's 44-18 defeat at Hull KR by dropping Jon Molloy and Vince Mellars for the Magic Weekend clash.
"We expect to get Ali back this week, Taulima has a chance of playing while Reece Lyne missed out last week so he's back in contention," Agar said.
"It's an understatement to say we were disappointed last week so a lot of these players have a strong chance of playing this weekend."