Liverpool's asking price of £35m remains some way above the improved offer, though it marks a clear step up from the roughly £21m proposal that was given short shrift last month.
This is far from Inter Milan's first attempt to land the midfielder — talks stretch back to a loan approach in January that Liverpool dismissed almost immediately given how late it came in the winter window, and Italian reporting elsewhere has tracked a series of rejected bids since, rising from an initial £17.3m through to the current offer relayed via intermediaries this week.
Jones has a year left on his contract and has been an Inter target throughout the year, though it remains unclear whether sporting director Richard Hughes and head coach Andoni Iraola would sanction a sale at this stage of the summer.
Iraola has already made his admiration for the 25-year-old clear during pre-season, telling reporters: "I rate Curtis very highly. For me, he is a great, great player and I hope he can continue with us and continue performing the way he has been performing. It's very important that he's Scouse, that he's from here.
"I also like the personality. From the outside at least he looks like a player with good character and I hope we can keep him, not only for this year but for more time."
Elsewhere at Anfield, a work-permit hurdle has forced a rethink over 18-year-old Ifeanyi Ndukwe, with the towering centre-back — Liverpool beat Inter to his signature back in January in a deal worth £2.6m — now set to spend the season out on loan instead of staying put.
The Austrian, who stands at 6ft 6in and captained Austria Wien's Under-16 side before reaching the final of the 2025 Under-17 World Cup, only became eligible to complete his move this summer upon turning 18, and had already caught the eye during pre-season: he replaced an injured Joe Gomez early in Liverpool's US tour opener, a 4-2 win over Sunderland, then went the distance in a 1-0 victory over Wrexham.
The original idea had been to keep Ndukwe on Merseyside through at least the first half of the campaign, but agent Georg Lederer confirmed that thinking has since been abandoned once it became clear a visa would not be forthcoming.
"The fact is that in the 2025/26 Premier League season, clubs had two exception spots where they didn't have to reach this points target," he explained to ORF. "In the new season, this extra rule will be reduced by one spot. A loan club will now be found where Ifeanyi can gain playing experience at a very high level."
Ndukwe's representatives at More than Sport by AS1 confirmed that several clubs have already registered concrete interest in a loan despite the setback.
"Despite this change, we see the situation as a great opportunity for Ifeanyi," they said. "Several top international clubs have already expressed concrete interest in a loan. Regardless of which path he chooses, he will have the opportunity to gain regular playing time at the highest level – a prospect that would not have been guaranteed to the same extent in his first few months at Liverpool.
"The original plan with Liverpool was to evaluate after the first six months whether Ifeanyi was getting enough playing time in the Premier League. Only then would a decision be made about a possible loan. It was always clearly defined that his development should take place at the professional level and not through appearances for the U21 team.
"Liverpool has clearly communicated to us that they are 100% committed to Ifeanyi and plan with him, because he has already demonstrated his potential, for example, in the friendly against Sunderland. These recent developments give Ifeanyi the opportunity to gain valuable experience at the highest international level, receive consistent playing time, and consistently advance his sporting development."