Netherlands international Van Dijk was forced off early in the Reds' 2-2 draw with Merseyside rival Everton on Sunday (AEDT) after a wild challenge from Jordan Pickford.
Scans have since revealed Van Dijk suffered damage to his anterior cruciate ligament, an injury that will ensure he misses the majority – if not the entirety – of the remainder of the season.
Liverpool have previously been linked with 21-year-old France international Upamecano and Carragher – the Reds' second-highest appearance maker – says Jurgen Klopp needs to bring him in as a matter of urgency when the January transfer window opens.
"Liverpool's next big signing had to be a centre-back anyway and I am talking next summer," Carragher told Sky Sports.
"Dejan Lovren went and they didn't replace him so they only have three centre-backs and one of those, Joel Matip, is always injured. There are question marks now about Joe Gomez.
"So, there is no doubt that Liverpool had to go big in the transfer market next summer. That was the next one. That has now got to be brought forward to January 1, whoever the target was in the summer. Not the end of January.
"There is a young man everyone was talking about at Leipzig, Dayot Upamecano. I think he is only 6'1" or 6'2", so he is maybe not that aerial but if he is the one they wanted then bring it forward. Pay extra money and do it on January 1.
"That means getting through 11 league games without Virgil van Dijk."