It has been widely reported that Liverpool have already had one bid for Coutinho rejected by Inter, with talks thought to be continuing over a deal for between £8 million and £10 million.
The Telegraph reports that Rodgers is in advanced negotiations with Inter and is hopeful that the 20-year-old will be a Liverpool player by the time the Reds play their FA Cup fourth-round tie at Oldham on Sunday.
Yet Coutinho has told Sky Italia that he remains keen to prove himself at Inter.
"The Liverpool offer? I'm happy with these offers, because it means I am doing my job well," the former Vasco da Gama man said.
"But I think Inter have faith in me and I want to show my value here. I don't feel that I'm on my way out. I hope to stay at Inter.
"The young players feel the faith of the coach and of the club: me, Juan Jesus, Benassi, Bessa and the others."
Supposedly more keen on a move is Liverpool goalkeeper Pepe Reina. Reports in Spain in recent days have suggested that Reina is one of the goalkeepers being considered by Barcelona to replace Victor Valdes, who has announced that he will not be signing a new deal at the Camp Nou.
And Reina’s father, Miguel—like Pepe, a former Barcelona goalkeeper—has said that his son would welcome an approach from the Catalan giants.
“I am absolutely convinced that Pepe would be more than delighted to return to his Barcelona, where they made him into an upstanding man from the time he arrived at the age of 12 until he left," he told Spanish radio station La Xerxa.
"I say it publicly, I would be delighted to see Pepe back at Barca, of course I would," he added.
"What father would not like to see his son at Barca? He has already been there but to return would be extraordinary."
It had already reported that Liverpool boss Rodgers has been eyeing up potential replacements for Reina. A new name has recently emerged to potentially add to that list in Mexico international Guillermo Ochoa.
The Daily Express has claimed that Liverpool have already offered Ligue 1 side Ajaccio £5 million for the 27-year-old.
Ochoa’s agent has suggested that if there is interest it is not yet at that advanced stage.
“Many clubs are interested in Memo [Ochoa], and they have been watching him, but we are still at the stage of rumors," Jorge Berlanga told French publication So Foot.
Meanwhile, there has been further speculation over the future of Liverpool forward Luis Suarez of late. Some reports have claimed that Pep Guardiola wants to take the Uruguayan to Bayern Munich for next season. Reds captain Steven Gerrard has intimated that the club may need to qualify for the Champions League in order to keep hold of the 25-year-old.
"Hopefully, we can put a fight on for him between now and the end of the season to finish fourth and we make a decision for him," Gerrard said of Suarez, according to Sky Sports.
"Only he knows what his ambitions and dreams are. Luis deserves to be in the Champions League - everyone has to admit that.
"Outside of Leo Messi and Cristiano Ronaldo, I'll argue he's as good as anyone else.
"He's probably the best player I've played with and I've played with some top players in my career."