The Swans look to have been priced out in their pursuit of the Italian international.
Tutto Mercato Web believes that Swansea have followed up on a previously unsuccessful €22 million (£16.7 million) bid for the player earlier in the window.
The new bid, of which the fee is undisclosed, has fallen short of Napoli's asking price once again and it seems Swansea are fighting a losing battle over the attacker. Napoli are said to value him at €35 million (£26.5 million) which would look well out of Swansea's price range.
Swansea are looking for a new striker after poor form from Bafetimbi Gomis, Eder and Marvin Emnes this season - with Andre Ayew being used down the middle more often than not this term.
New boss Francesco Guidolin is believed to be a big fan of his countryman Gabbiadini but he looks to be a far too expensive acquisition.
Gabbiadini has scored six goals in 16 appearances for Napoli this season and has six caps for the Italian national team - scoring once.