The Reds could decide to bringing in the Scot as a replacement for current Reds boss Rafael Benitez at the end of the season, according to the Mirror.
Benitez signed a new deal a year ago, that will keep him at the club until 2014, but Liverpool has disappointed both in the Premier League and Champions League this season.
McLeish is currently in contract talks with new Birmingham owner Carson Yeung as his current deal expires in 18 months.
