Klopp was appointed by Liverpool in October 2015 and extended his deal at the end of his maiden campaign, with another three years remaining on those terms.
The 52-year-old recently hinted that he may retire when his contract expires in 2022, but the European champion hopes to persuade him to stay on.
However, Kosicke has suggested Klopp is stalling on agreeing a fresh deal as he is not entirely happy with the weather on Merseyside – even alluding to the possibility global warming may impact the German's decision.
"Jurgen's contract is still valid until 2022 and it is an open secret that the club would like to extend," he said.
"We still have a little time for that. We must wait and see how everything develops and whether climate change will bring better weather to England or only to Germany.
"One should not underestimate [the bad weather]. I remember that in November or December, Liverpool first asked for a contract extension. I said: 'Let's wait'.
"Ulla [Klopp's wife] and Jurgen get up in the morning and it's dark. When they meet again in the evening, it is also dark or it is grey and there is drizzle.
"During the winter break in Germany, if the weather is really bad, the clubs go on vacation for two weeks, then come back and fly directly to the training camp in the sun for several days.
"During this time, the coaches in England sometimes have to prepare for up to 13 games. They are already exhausted and it's not so easy."
Klopp guided Liverpool to Champions League and UEFA Super Cup success in 2019, but outlasting Manchester City to win the Premier League is now a primary target.
"He is not only proud of the process that is going on, but also very happy that titles are back in the closet," said Kosicke. "Therefore, winning the Premier League would still be a goal for Jurgen.
"There are these six top teams in the league. Jurgen has recently said his biggest problem in Liverpool is Manchester City."