According to Honigstein and Hughes in The Athletic, Werner wants us:
This is the relevant bit, but the whole article is a nice one, with a bit of detail about Klopp's decision to stay:
"Klopp’s and Liverpool’s strong position is reflected in their stance on Timo Werner. After scoring the winner against Tottenham Hotspur in the Champions League last week, the RB Leipzig forward could not have made it any clearer that he would relish a move to Anfield if an offer came his way.
“I know that Liverpool is the best team at the moment in the world and when you are linked with that team it makes me very proud,” Werner said.
The Athletic understands Werner has already received proposals from Barcelona and Manchester United but is waiting to see whether Liverpool make a move. There is a clause in his contract which means that he can agree a £51 million move away as long as it is negotiated before a deadline in April, which would then give Leipzig longer to make their own plans to replace him for next season.
It was put to The Athletic by a key source this week that if Klopp chooses to go for Werner, who has 28 goals this season, negotiations are most likely to start in the middle of March. That is the month when discussions really got going in previous years on the eye-catching deals that brought in Fabinho and Alisson from Monaco and Roma respectively.
Immediately, Klopp is thinking about his attacking options for next season because of the scheduling of the Olympic Games (July 22 to August 😎 and Africa Cup of Nations (January 9 to February 6). Both tournaments might have an impact on a campaign that’s unlikely to involve either Adam Lallana and Xherdan Shaqiri, who are both expected to leave the club in the summer.
A longer-term consideration is that his brilliant front three are roughly the same age — Sadio Mane and Mohamed Salah both turn 28 in the next four months while Roberto Firmino is 29 in early October — and though Klopp wants them to finish their careers at Liverpool and achieve plenty more before then, the introduction of a younger forward (Werner is 24 next Friday) could be beneficial.
Werner would not be signed as anyone’s replacement but as competition, someone who has the ability to push one of Salah, Mane or Firmino out. Of the three players currently at Klopp’s disposal, only Firmino does not kick up a fuss when he is substituted — yet he is arguably the most valuable of them all. Even on Monday night against West Ham United, Mane seemed frustrated when he was taken off in stoppage time until a greeting from the Kop distracted him.
Mane and Salah have a healthy, competitive rivalry and Klopp benefits from this but he does not want to disrupt the balance. Loyalty is also a big deal to him. Both players signed for him and have given him their all. It will take conversation and careful reasoning if he is to convince each of his present front three of the benefits of heightened competition at a point in the calendar where promises of the earth from elsewhere will probably come to their agents. The timing of such conversations will also be crucial considering what Liverpool can win from here.
Werner, meanwhile, is said to be perceptive. He has seen the struggles of Philippe Coutinho after a big move from under Klopp’s Anfield wing to Barcelona and knows that club’s current team is probably its weakest since before Frank Rijkaard took charge in 2003. He also knows Manchester United have been in flux since 2016.
It has been suggested Werner also does not want to waste prime years waiting for Liverpool’s star players to move on, but surely Klopp has noted another comment from last week where he admitted, “I know that Liverpool play a lot of good players and I have to improve myself and I need to learn many more things to get to that level and to play there.”
Those who have followed Werner’s career closely in Germany say his character is a mature mix of talent, patience and perseverance. He is strong-willed. There had been an agreement for him to go to Bayern last summer but they messed him around by pursuing Manchester City’s Leroy Sane more aggressively instead. When that move broke down in August because of a serious injury to Sane, it was too late to turn back. Werner was said to be so disappointed by the way Bayern handled the matter that he instructed his representatives to return the contract — an indication it would never be signed.
He has gone from strength to strength this season, emerging as one of Europe’s best forwards, and is still wanted by a section of Bayern’s board, who were banking on his original Leipzig deal running out so they could sign him on a free transfer this summer. This left Werner feeling like he wasn’t a priority — or, at worst, not really wanted — so he agreed a new contract which gave him freedoms around fresh options over the next few months."
https://theathletic.co.uk/1624500/2020/02/27/timo-werner-liverpool-barcelona-manchester-united-transfer/