The Bayern Munich winger has scored three goals for his country at the 2014 World Cup, with Louis van Gaal's side set to face Costa Rica in their quarter-final on Saturday.
And Carragher says Rodgers made a move to sign the star last year, at a time when the 30-year-old was struggling for game time at Bayern – with the defender claiming such a move would have been perfect for both parties.
"Brendan Rodgers tried to sign him for Liverpool before the 2013 Champions League final,” he told the Daily Mail.
"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."
Former Chelsea and Real Madrid star Robben signed a new contract with Bayern earlier this year, which would keep him at the Allianz Arena until 2017.