French midfielder Vieira was adored at Arsenal, where he led the club to three Premier League titles.
The former Arsenal captain was ready to play for Tottenham, though, despite the animosity between the two north London rivals.
Redknapp thought it was an 'incredible decision' but insisted that if anyone could cope with the expected abuse from Tottenham fans, it was Vieira.
The move never eventuated but even the prospect of Vieira wearing Tottenham's white is sure to anger Arsenal fans.
Redknapp's new book, A Man Walks On To A Pitch, is being serialised by the Daily Mail's Sportsmail. A excerpt reads: "Patrick Vieira was a different type of midfield player and one every club would like to take now.
"He almost came to me at Tottenham from Inter Milan.
"I thought it was an incredible decision by him, after he'd been such a hero at Arsenal.
"I met him at his house in Hampstead and he had no fear about what people would think or what reception he would get. He had enough confidence in his ability to just brush it away.
"I remember talking to Daniel Levy, our chairman, about him. He said the crowd would be hostile.
"I knew that - but if he had enough bottle to want to put on our white shirt after all that had gone before, I thought it said something about the man.
"In the end, his circumstances changed and he decided to stay in Italy.
"By the time he did become available - the following January - our needs had changed and Manchester City snapped him up."
Redknapp also said that one of his biggest regrets in management was not signing Luis Suarez from Ajax when he had the chance at Spurs.
"Despite all that has happened in his career since, one of the biggest regrets of my life in management is not taking Luis Suarez to Tottenham when we had the chance," he wrote.
"'You've got to take him, Harry,' Ruud Gullit told me. 'He's fantastic.'
"Yet all of our scouts' reports said that at Ajax he played wide off a striker, whereas we were looking for a proper frontman.
"Just when I was on the verge of taking the plunge, the price went up. In the end he was sold for £22.8 million to Liverpool. We wouldn't have paid that."
