"I was talking to Steve McClaren and I asked him jokingly about the new captain. He smiled and he did not say anything," said the Reds' boss.
"For sure, it would be really good for Steve. He's more mature now and he could do both jobs, captaining Liverpool and captaining England.
"Maybe it would help him in terms of really, really focusing on football and show him how important it is to be captain.
"As captain he is an example to the rest of the players on and off the pitch, but his main quality is the quality of his passing on the pitch.
"I think he's learning things and he's thinking more about the game. As a manager, you always want to see your players improve and I think Steve has the potential to improve more.
"Each year I can see him adapting more to the game and he understands better what's going on. He can analyse the situation and try to do the right thing.
"He's a good player at a good club and that means he can fight for trophies. He's playing in big games and he's matured more because of these big games.
"If you're an ordinary team, normally you only play in the Premier League and in some cup competitions. You don't have the possibility of playing for trophies.
"In the last two years, he's had those possibilities and he's experienced a lot of different games."