Reina and Carragher, both mainstays of the Reds rearguard since the Spanish shot-stopper joined the club in 2005, clashed in an unnamed game because of a disagreement about passing options.
"The veins in our necks were bulging, our faces were red with anger and neither of us were going to back down," Reina said in his autobiography Pepe, which is being serialised in the Daily Mirror.
"The row between me and Jamie Carragher started during the game - I can't even remember who we were playing against - carried on in the dressing room and didn't come to an end until we sent each other text messages apologising.
"He went crazy at me, screaming at me, and I was screaming back at him. We were like that for fully five minutes shouting at one another and getting our point across. When we got back into the dressing room the row started again and it was even rougher than it had been on the pitch.
"The other players were just totally silent, watching us giving each other hell. It wasn't even over anything major. It was just a difference of opinion over whether I should knock the ball long or [if] he should give me an angle for a short pass.
"I ended up kicking the ball out wide to where I thought Carra should be, but he wasn't there and it went out of play. He shouted at me: 'You really think you're Franz Beckenbauer and you're always trying to play short passes when you should just empty it.'
"I responded: 'If I know one thing it's that you definitely aren't Beckenbauer, but just give me a bit of support when I've got the ball so we can try and play instead of just kicking it down the pitch.'"
Meanwhile, Reina has lifted the lid on an occasion when he was at loggerheads with former Liverpool boss Rafa Benitez over alcohol consumption. "The biggest argument I had with Rafa was about a beer," Reina said. "We had lost 2-1 to Besiktas and not played too well. I decided to have a beer. I don't see a problem with that because for me it is healthier to do that than it is to have a glass of Coca-Cola or a drink like that.
"Suddenly, the bartender turned up and said: 'The club doctor, medical staff and manager have made it clear to me they do not want you to have a beer.' I still don't know why this was the case but whatever the reason, I was fuming. I grabbed my bags and stormed off to my room without even having dinner. That is how upset I was.
"For me, Rafa was in the wrong. It suggests the manager does not trust you to look after yourself in the right way. He saw it differently and, if he has a small weakness as a manager, it is that he did not have a massive career as a player before he moved into management."