Former Liverpool defender Carragher suggested earlier this season that Jurgen Klopp should look to sell Sturridge this summer.
And Sturridge, 27, wanted Carragher to clarify his comments.
The England international scored seven goals in his 27 appearances for the Reds this season, but managed just eight starts in the Premier League.
And Carragher told Sturridge a player of his quality should not be sitting on the bench, especially if the club could get £25million from his sale.
Daniel Sturridge not happy with Jamie Carragher's transfer comments
Daniel Sturridge unhappy with Jamie Carragher's transfer comments
"I've got a funny story about that, and I didn't mind it actually – I was happy about it," he told the Liverpool Echo.
"As soon as I got on the plane, I could see Sturridge wasn't his normal self with me.
"And he pulled me in Australia after the game, or at half time when we'd come off.
Jamie Carragher says Daniel Sturridge is too good for the bench Jurgen Klopp enjoying Australia on Liverpool tour Thu, May 25, 2017
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Jurgen Klopp is having a great time in Australia for Liverpool's pre-season tour
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"He said, 'why did you say they should sell me?'. I said, 'oh, fair enough!'. I went straight back and said, 'well, what else can they do?'!
"He wasn't too happy that I'd said that, but my point was that a player of his quality, you either play or you go.
"If I was him, I wouldn't want to be on the bench, and from [Jurgen] Klopp's point of view there is nobody else on the bench he could get money for, if he needed say £20m or £25m."