The Reds are rumoured to have secured a deal for Chelsea front-man Daniel Sturridge, but manager Brendan Rodgers is still thought to be keen to add further attacking options.
The Latics are hopeful that they can keep hold of the 23-year-old, but are fearful that he could leave on a free once his contract expires at the end of the season.
This may tempt them to cash in in January, with Di Santo’s stock at a real high.
The forward found himself called up to the Argentinian national team in October for their friendly with Saudi Arabia, after impressing coach Alejandro Sabella.
Rodgers is an admirer of the front-man, whom he coached when at Chelsea, and will be confident of persuading him to swap the DW Stadium for Anfield.
A January move could be a possibility, but if he is unable to agree a fee the Northern Irishman may attempt to negotiate a deal to bring him to Merseyside for free next summer.
It appears likely that Andy Carroll will leave the club on a permanent basis at the end of the season, which will tempt the Merseysiders to draft in another forward player.
Wigan may attempt to bring in Leeds striker Luciano Becchio, who has netted 17 goals this term, if they are unable to persuade Di Santo to stay.