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Reds' No.1 goalkeeper Alisson Becker is sidelined with injuries and deputy Adrian San Miguel was woeful in the 7-2 thrashing at the hands of Aston Villa before the international break.
And Murphy believes it's in Klopp's best interests to move for a goalkeeper, specifically Butland.
Capped nine times by England, the 27-year-old is out of favour at Championship outfit Stoke and has not played since July when he was displaced by Welsh shot-stopper Adam Davies.
Liverpool would need to snap him up before the domestic transfer window closes at 5pm on Friday but Murphy reckons Butland would be a "clever" addition.
"I like the thought of Jack Butland joining Liverpool," Murphy told the Football Index Podcast.
"He is trying to find himself again after a really good start to his career when he was being spoken about as potentially England's first choice goalkeeper.
"Unfortunately for him, he's fallen by the wayside a little bit with Stoke being relegated from the Premier League.
"This might be a tremendous opportunity for him to go to a wonderful club, learn from a brilliant goalkeeper, play some games, and start again. That makes sense to me - I think signing Jack Butland would be a clever bit of business."
It's been reported that the Premier League champions have been in talks with Butland, whose contract at the Bet365 stadium is due to run out in 12 months.
Alisson is still recovering from a shoulder injury suffered in training and Klopp has not put a timeframe on when he expects his first-choice keeper to return.
"Yesterday an incident in training, very unlucky - two players clash with each other, one gets up, one doesn't. The one who didn't get up was Ali," he told Sky Sports prior to the shock result at Villa Park.
"So, after the first few things, we don't know 100 per cent. But he cannot play today. We have to do further assessment; but he's injured, let me say it like this. Now we hope it will go really quickly.
"It's an international break, I'm pretty sure after that he will not be ready. Then we have to see week by week."
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