The rejection was reported by journalist Ben Jacobs, who says Jones will be going nowhere unless a suitor is willing to put the €40m Liverpool are holding out for on the table.
Inter Milan's fascination with the 25-year-old is nothing new. The Italians first tested the water in the closing stages of the January window, only for Liverpool to make clear the academy graduate was not for sale mid-season.
The pursuit resumed this summer, and Inter have now failed with three separate offers. An opening bid of €20m was swiftly dismissed, and a second worth €25m fared no better, with Liverpool reportedly unimpressed by what they saw as a less than serious approach from the San Siro club.
The €32m proposal represents another significant jump, but it still leaves a gap of €8m to Liverpool's asking price, and the Anfield hierarchy have shown no sign of blinking.
Time may yet become a factor in the saga. Jones' contract runs until the summer of 2027, meaning he is entering the final 12 months of his deal, and Liverpool would risk seeing his value erode if he enters next season unsold and unextended.
For now, though, the club appear relaxed. Jones has been a valuable squad member across recent campaigns, and his homegrown status adds further weight to the case for keeping him as new head coach Andoni Iraola takes charge of pre-season on Merseyside.
Jones is not short of admirers elsewhere either, with Nottingham Forest among the clubs credited with interest in recent weeks, and the England international is understood to be open to a move in search of more regular first-team football.
Whether Inter return a fourth time remains to be seen. The Nerazzurri have shown remarkable persistence in the face of repeated rejection, but each knock-back has only hardened Liverpool's stance, and the message from Anfield is unambiguous: meet the €40m valuation or move on to other targets.