Liverpool were interested in Southampton star Van Dijk throughout the summer but Saints refused to sell the Dutchman.
He handed in a transfer request but Saints rejected it and forced him to stay at the club, just over a year after signing a new long-term contract.
Given says there should have been a Plan B for Liverpool boss Klopp, whose defence has come under severe criticism this season to date.
“It was obviously well-documented they wanted Van Dijk in the summer,” Given said on Sky Sports.
“He didn’t arrive for whatever reason but you would have thought a club the size of Liverpool would have had a plan B.
“Obviously Van Dijk on the last day, we kind of knew that it wasn’t going to happen.
“You think a club of that size would say ‘well we can’t get him, we tried everything’, so there should be a Plan B, there should be another centre-half in Europe or world football that they could get to their football club.
“It’s clear the goals from last season and this season they’re conceding, and even Klopp himself is saying he’s sick of the game goals week in, week out.”
Given says Jurgen Klopp must work hard with his current crop of defenders on the training ground as he would need a “magic wand” to bring someone in outside the transfer window.
“You have to (work on it) now because the window is closed, there is no magic wand,” the former Newcastle goalkeeper added.
“Van Dijk couldn’t come in or the like, so it’s something they have to work on. Players have to step up and just be a bit more dominant.
“It always seems at Liverpool that someone else is going to head it, someone else is going to clear it.
“No one ever seems to take control of the situation. You can see how frustrated he was at the end of the game.
“You can have all the possession and maybe they’re not putting the chances away as well but as you say when you’re conceding at the back, you’re always going to have to score extra goals at the other end.”
