The England midfielder is out of contract at current club Manchester City and is being heavily linked with a move to Brendan Rodgers’ side, with BBC Sport suggesting they have secured his signature.
The Reds are of course losing captain Steven Gerrard as he departs for LA Galaxy, and Carragher believes Milner could provide competition to current vice-captain Jordan Henderson for the captain’s armband.
“The fact that Brendan Rodgers wouldn't confirm Jordan Henderson as this season’s captain makes me think him and Milner could be competing for the armband,” he said.
“Was that another thing that enticed Milner to join? I think it’s a good signing and I’m glad he’ll be wearing the red of LFC hopefully when it’s confirmed in the next week or so.”
Carragher has also discussed where the England midfielder will fit in at Liverpool, suggesting he will not only compete with Henderson for the captain’s armband but for his position as well.
Milner has often been used in a wide midfield role at City but is arguably at his best in the centre, where he has impressed in the past, but is a position he has failed to break into at the Etihad.
And Carragher says he has no doubt that is why Milner is joining Liverpool, to play more centrally, but it will be interesting to see how he could fit in.
“I have no doubt one reason he's joining is to play in a more central role rather than out wide,” he added.
“His best position is tucked in from the right in a central role which is also Jordan Henderson's so it'll be interesting how those two fit into next year’s midfield set up.”