Former French international and World Cup winner Vieira became a true Gunners icon during a nine-year stint with the club between 1996-2005, winning three Premier League titles, four FA Cups and captaining Arsene Wenger's side following the retirement of long-term skipper Tony Adams.
The 38-year-old eventually left Arsenal for Juventus after a lengthy pursuit on behalf of the Serie A giants before moving on again to Inter just a year later.
It was during that aforementioned stint at the San Siro that Redknapp claims he almost snared Vieira, with the midfield enforcer apparently not concerned about the potential reaction from the Arsenal supporters that obviously held him in such high regard for his commitment, leadership and sheer tenacity on the pitch.
"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," the experienced boss writes in his new book - entitled A Man Walks On To a Pitch - that is currently being serialised by the Daily Mail.
"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."
"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."
In a further point on Vieira, Redknapp - clearly in awe of the player's character and ability - questions why Arsenal did not try to bring him back to the club following his decision to retire in 2011.
"Patrick is the player Arsenal have never really replaced, but I still find it bizarre that he ended up in a blazer at Manchester City. Shouldn’t he be at Arsenal?"
Vieira - who has also impressed as a television pundit in recent years -remains at Manchester City three years after hanging up his boots, initially cutting his coaching teeth in youth football before swiftly making his way up to become the head of elite development at the Etihad Stadium.