According to La Gazzetta dello Sport, Inter Milan's hierarchy at Viale della Liberazione want to push the deal through after an initial attempt in January came to nothing.
Liverpool are seeking €30 million for the 25-year-old midfielder, while Inter's last offer stopped at €20 million.
A new, improved bid is now being prepared, according to the same report.
Jones came through Liverpool's academy after joining the club at the age of nine, and had never previously considered leaving his home city before Inter made their interest known.
His contract situation has added urgency to Inter's pursuit. Jones is under contract at Anfield until 2027, meaning he could agree pre-contract terms with another club as early as January if no transfer is completed this summer.
That scenario is one Liverpool are determined to avoid, given that allowing Jones to leave for nothing in twelve months' time would deprive the club of a transfer fee they could otherwise secure now.
Inter Milan manager Cristian Chivu has made clear he sees Jones as an ideal fit for his midfield, describing the England international as a versatile profile the Nerazzurri currently lack.
Jones made 34 league appearances for Liverpool last season, though only 18 of those came from the start, with the academy graduate also deployed at right-back during a stretch of the campaign when injuries forced Liverpool to improvise.
According to The Athletic, Liverpool do not want to lose Jones but would reluctantly sanction his departure if the midfielder pushes for a move and a suitable offer arrives.
Inter sporting director Piero Ausilio has been open about the club's interest in previous public comments.
"Curtis Jones, we are paying attention to him. We didn't hide it. We understand what the developments will be."
Liverpool's valuation of Jones is understood to be informed in part by Conor Gallagher's January move from Chelsea to Atletico Madrid, which involved a fee in a similar range.
Reports in Italy suggest the two clubs could ultimately settle somewhere in the region of €25 million, a figure that would split the gap between their respective positions.
For Inter, signing Jones would give Chivu a multi-functional midfielder capable of contributing in several roles, at a moment when the club is also working through other midfield business, including the uncertain situation surrounding Davide Frattesi.
With both sides reportedly eager to avoid a drawn-out saga, the next move belongs to Inter — and how much more they are willing to put on the table will likely determine whether this transfer finally gets done.