The defender moved to the Scottish side from Ipswich Town in the summer of 1986, despite United's interest.
Butcher, who joined Rangers under Graeme Souness, said the Ibrox side secured his services because they made a cash bid.
He said on the William Hill SPFL's YouTube show, The Warm-Up: "It was the Mexico World Cup that summer and I was told to go out there and do my best.
"Chris Woods was there [at Rangers] and there was also a journalist who was helping Souness put the Rangers team together.
"First Colin West signed, then Chris Woods, and then I was told ‘the gaffer wants to talk to you, he wants to see you’.
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"I was like ‘wow’, but I had a job in hand and a World Cup to play for.
"It was when I went back to England that Rangers came back in with a cash bid.
"Most clubs back then paid in bits and pieces, but it was cash and Ipswich were happy."
Butcher also revealed that Manchester United tried to sign him on three occasions.
He said: "I am not being big-headed but prior to the move to Rangers, Man United had come in for me three times.
"I knew where I was going to stay, I knew what schools the kids were going to, Brian Robson would also tap me up, so I thought it was going to get done.
"I waited and waited, but it never happened.
"Then Souness came in and bang, we were agreed and done.
"It was the start of the revolution and if Souness wants you, you’ve got to go.
"When it’s that real high standard of decision making, when someone like that wants you, you have to go.
"I’d also spent half my career in England trying to kick him and ‘if you can’t beat them, join them!’"
Terry Butcher appeared with Kris Boyd and Steven Naismith on The Warm-Up, the William Hill SPFL’s weekly preview show. Check out the full episode HERE.