Virgil van Dijk and Joe Gomez are both set for long periods in the treatment room, while Trent Alexander-Arnold and Fabinho have also struggled with injuries this season.
Austria international Alaba has made close to 400 appearances for Bayern Munich, helping the club win nine Bundesliga titles and establishing himself as one of the best players in Europe.
The 28-year-old is coming to the end of his current contract and has so far rebuffed offers to extend his stay in Bavaria.
Asked whether Liverpool should attempt to sign Alaba, former England striker Heskey told Rousing the Kop: ‘I think so.
‘You’ve got a player who plays left back, left side centre back and left wing back and he can step into midfield as well.
‘You’ve got a younger version of James Milner, someone who can tick all the boxes but tick them well.
‘If you look at what he’s achieved, he’s a winner, so he knows what it takes to win and that’s what you want within your squad.
‘Because that rubs off on the younger players as well because what you do and what you’ve achieved, they want to do as well.’
Fellow former Liverpool star Danny Murphy has also backed Klopp to sanction a deal for Alaba.
He told Stadium Astro: ‘In terms of his ability, he absolutely fits, he’s a super player.
‘I suppose what you’d have to be looking at from Klopp’s perspective is the balance of the squad moving forward when everyone is fit.
‘You can’t have enough good players, but I think there would be a lot of competition.
‘He would certainly add quality and strength in depth, but there will a lot of takers for Alaba.’