Bayern Munich's Thiago Alcantara would love to play for Jurgen Klopp, reports Kaveh Solhekol, but Liverpool are not currently looking to strengthen in midfield.
There is no indication that situation will change either but, if it does, the 29-year-old Spain international would be a name under consideration.
Thiago has one year left on his deal at the Allianz Arena and he has told them he wants to leave for a new challenge.
Bayern's sporting director Karl-Heinz Rummenigge said this week they will accept a "reasonable price" for him too - believed to be £30m.
But in Fabinho, Naby Keita, Georginio Wijnaldum, Jordan Henderson, Alex Oxlade-Chamberlain, James Milner and Curtis Jones coming through, Liverpool have lots of midfield option as things stand.
Flick: Thiago should go to the Premier League
Bayern Munich head coach Hansi Flick feels a new challenge in the Premier League would be ideal for Thiago.
In an interview with Sport Bild, Flick said: "Thiago played for Barcelona and seven years for Bayern Munich. He wants a new challenge which, in my opinion, should be the Premier League.
"He is a very important player for us, who gives the team that certain something, and we have to replace him with the same quality."
Flick's comments echo those of chief executive Karl-Heinz Rumenigge to the Munchen Abendzeitung on Tuesday.
He said: "For a while it looked like he wanted to stay but at some point he obviously thought that he would like to do something different at the end of his career.
"Thiago is 29 now and as soon as the "3" is up it will be a little more difficult to find a top club in Europe, so you have to understand him a little bit. He had a top club in Spain, a top club in Germany so maybe he would like to join a top club in England."
Thiago could be the missing piece in Liverpool's jigsaw were the Bayern Munich attacking midfielder to move to Anfield this summer, German football expert Raphael Honigstein told the Transfer Talk podcast.
"In Germany they are saying that if Liverpool buy Thiago, they'll win every game next year!" he said.
"He's just been so good. If there's one player they're missing in that midfield it's that technical player. More of a creator and a dribbler, somebody who scores spectacular goals and can find a pass.
"It's not a player that Liverpool have accommodated in recent years and it would change the dynamic of their midfield if you find space for him.
"Bayern had an offer of a contract extension on the table. They thought they'd agreed, but he's changed his mind. The club assumed that he must already have a deal in place with someone else, but that side are yet to reveal their hand.
"There's been no firm offer yet from anyone, but Bayern are bracing themselves for Liverpool because that's the rumour in the dressing room. But we haven't seen any offer yet and Liverpool have distanced themselves from making a move for Thiago. That could just be part of their strategy, but Bayern are more or less resigned to him leaving.
"I think that would be a big blow for them as he's more or less been running games for them and is an absolute key player in my mind. It'll be very difficult for Bayern to find a player who is like Thiago, who's so unique in his technical ability. It's a blow for Bayern and something that they didn't expect."
Wijnaldum's agents waiting for contract talks
Wijnaldum's representatives are waiting for Liverpool to open talks about a new contract, however, as he enters the final 12 months or his current one.
The Netherlands international, who joined from Newcastle in 2016, loves playing for the club and there is no suggestion the 29-year-old is going anywhere this summer.
He has won the Premier League and the Champions League at Anfield after joining for £25m and there is no reason he cannot keep winning more trophies with Klopp's current side.
Analysis by Sky Sports News' Liverpool reporter Vinny O'Connor:
"Liverpool have to improve because if you look at the likes of Chelsea at the moment, Man City are going to improve, Man Utd have had a run that has taken them into the Champions League once again.
"So Liverpool know they have to be better next season and if they are not going to do so much in the transfer market, then the Liverpool squad know they will have to raise their levels once again.
"Klopp has said they are not necessarily going to do too much business in the transfer window, although there could be the odd addition. But then there are the other improvements to the players they have already got. We saw with Naby Keita's performance, if you can get those kind of performances out of him consistently, that is an improvement Liverpool could make.
"And Klopp talked to us after the game about Curtis Jones and he could not help let it slip that he is ready now, they are very excited about what he can produce for Liverpool FC. Adam Lallana has left a space and Jones can more than fill that and become an influential member of that squad as there is a lot of excitement at what he can achieve in a red shirt. And he is taking on the No 17 shirt next season, which of course in the early years was Steven Gerrard's."
All the latest news, analysis and rumours from the window in one place.
The summer transfer window will run for 10 weeks from July 27 and close at 11pm on October 5.
A domestic-only window for trades between the Premier League and EFL then runs from October 5 and closes on Friday October 16 at 5pm. Follow all the news and analysis on Sky Sports News and across Sky Sports' digital platforms, including with our dedicated Transfer Centre blog.