James Maddison was the subject of a second £50m offer from the Tynesiders.
The Magpies are waiting for a response to their improved offer for the 25-year-old.
The value of Maddison is thought to be close to $60m.
The club's record signing would be Maddison if the offer was accepted.
Despite setbacks in the transfer market this summer, Eddie Howe still wants to strengthen his squad, even though they were unsuccessful in their attempts to sign Anthony Gordon and Hugo Ekke.
Three summer signings have been made so far, with the arrivals of Matt Targett, Nick Pope and Sven Botman.
Brendan Rodgers said after the win over Sevilla that he wanted to keep Maddison.
Rodgers insisted that they won't be taken advantage of over Maddison or others, despite the fact that they are the only top-flight side yet to make a signing.
The club's financial situation isn't ideal in terms of wanting to do business. He said that it doesn't mean that we will be exploited and that we don't want to lose our best players.
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You don't want to lose your best players and not be able to find replacements. That doesn't match up.
He is the best James has been. He's a pleasure to work with. I first came in to a 25-year-old father and I loved him as a young man.
He is the key to our success. I don't think I'd want to sell him.
A deal for the French defender is being worked on, with Rodgers issuing a warning over the matter.
Fofana has five years left on his contract and Rodgers is adamant that he won't be leaving.
He denied that he was for sale. He had 17 league starts. He flourished and developed not just as a player but as a young man after everyone had a look at him.
I have no doubt that in the future he will be on that stage but he is very comfortable here.
He has a great relationship with his team-mates, which is important, and there's no doubt he will perform well on that stage. He's not for sale at the moment, he's developing.
He has an opportunity to get himself in the France squad for the World Cup and I'm pretty calm on that. I've talked to him many times.
He can't stop the speculation, but he's concentrating on his work and development.
The transfer window will close at 11pm on September 1st.
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