The ex-Liverpool defender claims the Northern Irishman had targeted Robben before the 2013 Champions League final, only for the Dutchman to score the winning goal for Bayern Munich and go on to stake his claim within the German side.
And Carragher believes Robben would have been the Reds' first world-class winger since John Barnes had he signed.
"Brendan Rodgers tried to sign him for Liverpool before the 2013 Champions League final," Carragher said.
"He wasn't a regular for Bayern then but he scored the winner at Wembley against Borussia Dortmund and has since gone from strength to strength.
"Liverpool have not had a world class wide man since John Barnes."
The 30-year-old is now reportedly a target for Louis van Gaal at Manchester United and has been instrumental in the Netherlands' run to the World Cup quarter-finals. Just imagine the difference he could have made, then, had Rodgers sealed the deal and brought the Dutchman to Anfield.
As stats from WhoScored.com show, Robben scored 11 times and registered six assists for Bayern in the Bundesliga last term, averaging three dribbles per game and 1.4 key passes per match.
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Already, Opta stats prove how valuable Robben would have been for the Reds based on his displays in Germany. No Liverpool player averaged more won dribbles per game than the Dutchman, while only their three most successful players scored and assisted more frequently.
Raheem Sterling's form proved a huge bonus for the Anfield side in 2014 - but imagine the experienced Robben either standing in for the 19-year-old or playing alongside him in an attack that could terrify the best of defences. The Bayern man could have fit into Liverpool's 4-3-3 system or played at the head of their diamond formation, with Rodgers' men having already produced a goalscoring glut last campaign.
Towards the end of the season, meanwhile, Robben could have made a vital difference, with the Reds' squad being threadbare as they headed into crucial clashes against Norwich City, Chelsea and Crystal Palace. The likes of Joe Allen and Lucas had to be deployed in attacking midfield roles during stages in those fixtures - and the Dutchman's presence would have ensured that didn't happen.
Given that Liverpool finished just two points behind Manchester City in the Premier League title race, then, history could have told the story a lot differently had Rodgers managed to recruit the 30-year-old.