The Reds face Stoke City over two legs in Capital One Cup semi-final as the Merseyside outfit bid to win the competition for a record ninth time.
Liverpool make the trip to the Britannia Stadium for the first leg fresh from a humiliating 2-0 loss to West Ham United at Upton Park on Saturday afternoon.
Former Republic of Ireland international Kilbane explained why winning the League Cup would be huge in the context of Klopp’s Liverpool reign.
“He can win a trophy and get his first one inside three or four months,” Kilbane told BBC Sport.
“When Brendan Rodgers took over from Kenny Dalglish, Kenny had won that trophy. He did well to win it! This could be a big season for Jurgen Klopp to get that first trophy under your belt and kick on for the next few years.
“It would be a big statement to make.”
Liverpool are in eighth place in the Premier League table and 12 points behind leaders Arsenal following Saturday’s disappointing result at Upton Park.
The Reds have lost four of their 12 Premier League games under Klopp after the former Borussia Dortmund manager replaced Brendan Rodgers following October’s Merseyside derby.
Liverpool were 1-0 winners at Stoke on the opening weekend of the Premier League.
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