The correspondence reveals that Sony - which was reportedly already not happy with Garfield - might give up the franchise to comic book giant Marvel.
It is believed that Marvel would prefer to start the series from scratch, putting Garfield's role as Peter Parker, which he has played in two movies, in jeopardy.
The 30-year-old has played the role of the superhero in the 2012 and 2014 releases of the series, alongside Emma Stone as love interest Gwen Stacy.
If Marvel does take over the production of The Amazing Spider-Man, as suggested in the emails revealed by hackers Guardians of Peace, then it is possible both stars could face the axe.
According to Latino Review, if the takeover goes ahead then the new films will take a different turn, focusing on Peter's difficult childhood dealing with his powers, as well as romance side-story instead of being just an action film.
There have also been suggestions that the next Spider-Man film will feature a female lead, but it is understood that Marvel does not support this idea.
Plans for an Aunt May spin-off are also said to have been shelved.
The Spider-Man emails are just the latest in a series of Sony correspondence revealed by hackers.
Other new developments include an email from Columbia Pictures executive Michael De Luca to Sony co-chairman Amy Pascal, suggesting that American Hustle director David O. Russell was caught 'feeling up' his transgender niece's breasts.
The email said: 'David’s a loon, talented, but geez once I saw him bring Sally Field to a party and reduce her to tears. Plus remember when he got in trouble for feeling up his transgender nieces [sic] boobs?'
The incident, which went public in 2012, did not result in criminal charges after Florida sheriffs investigated.Earlier hacks featured racist emails mocking President Obama, and insults directed at Will Smith's children.
Most of the exchanges involve Sony Pictures chairman Amy Pascal, who is now said to be fighting to keep her high-powered job.
Other disclosures included accusations that Leonardo DiCaprio is 'despicable' for pulling out of a film, and a description of Angelina Jolie as a 'minimally talented spoiled brat'.
Further disclosures have centered on the upcoming release of The Interview, a film depicting the assassination of North Korean dictator Kim Jong-un.
Exchanges reveal pressure was applied to director and star Seth Rogen to change the death scene, with some suggesting the hackers are linked to the North Korean regime.
George Clooney has also been sucked in to the affair after emails emerged showing he was mortified by the failure of 2014 film The Monuments Men, which he co-wrote.
In an email to Ms Pascal, Clooney begged for 'protection from all the reviews' as critics began to widely pan the Second World War drama.
'Let's just make it a hit. I haven't slept in 30 hours. And it's 7 am,' he said.
Emails from Channing Tatum and the father of Beyonce have also been leaked, with posts online suggesting the hackers are set to reveal even more internal correspondence before Christmas.
Screenwriter Aaron Sorkin condemned the releases, saying news outlets who published details taken from hacked emails as 'morally treasonous and spectacularly dishonorable'.
He also denied accusations stemming from an email sent by Ms Pascal claiming that he was 'broke' and having an affair with a woman whose book he was hoping to use as a source for his next screenplay.
