Klopp said recently that “the club is in talks with Milly" over extending his stay with Liverpool, but he has now used his programme notes ahead of Wednesday's win against Leeds to clarify the situation.
“It's no secret that James is in the last year of his current deal – and again – I emphasise 'current'," he wrote.
“I spoke recently about 'talks' taking place to extend his stay with us and I thought it might make sense to use this platform to clarify that as I've seen some misleading reporting around it.
“By talks I mean I indicated my wish that James stays with us beyond this season, in a playing capacity. James will keep playing, he has told me this. So he should, by the way.
“The level he is at, physically, technically and in terms of personal desire and hunger for more, means he should play on for as long as that lasts.
“I hope it will be here and that is what I have told those at the club who take on these tasks.
“It's not fair or right to then jump to the conclusion that an offer has been made, because I don't think that's fair on James.
“I know I don't normally go into this much detail on matters such as this, but given how conscientious James is when it comes to not putting private things into the public domain, I wanted to make the actual position clear.
“As should be the case, these things will get picked up in private and James will keep doing for us what he has done during my entire time working with him.
“He gives his all, he helps the collective and he sets the standards. We are all very blessed to have him at our club."
Milner has made 275 appearances in all competitions for Liverpool – the vast majority of which have come under Klopp, who arrived at the club in October 2015.
“What he does for us is so valuable," Klopp added in the programme.
“And I make a point of keeping my language in the present tense because he continues to do it, day in and day out.
“He has been a constant during my time at LFC and I was thinking just the other day that I can't imagine what it would have been like to try and do all this without him here."
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