The Birmingham City manager reportedly spoke with the forward at the match between Tottenham Hotspur and Manchester United at White Hart Lane last weekend.
The 36-year-old former Leeds United and Liverpool forward is a free agent at the moment, having left Major League Soccer outfit Los Angeles Galaxy in December.
According to The Mirror, Redknapp, who worked with Keane at Tottenham, spoke with the former Republic of Ireland international last weekend.
The duo attended Tottenham Hotspur’s Premier League match against Manchester United last weekend in what was the North London club’s final game at White Hart Lane.
Keane was a superb forward at his peak, but at 36 would he able to score regularly in the Championship?
Birmingham narrowly avoided relegation to League One this season, but during the 2017-18 campaign the Blues will be aiming to finish in the top six of the Championship.