His Instagram video sees him stare into the camera, slowly lifting and then lowering a boxed Superman doll, while The Blue Danube which features barking dogs plays in the background.
It follows The Hollywood Reporter saying the British actor is parting ways with Warner Bros.
Warner has not confirmed the story.
While no decisions have been made regarding any upcoming Superman films, weve always had great respect for and a great relationship with Henry Cavill, and that remains unchanged, they said in a statement to CNN.
Anna Smith, film critic and president of the Critics Circle, told the BBC that if the speculation about Cavill leaving the role is true, shed love to imagine its because he is preparing take on the role of James Bond.
She added: The DC movies have been squandering his gift for suave comedy, which we saw in The Man from U.N.C.L.E. and which would take Bond back to the old school (minus the sexism, I hope).
What, if anything, happened here?
Hollywood Reporter suggested negotiations with Cavill broke down over scheduling problems for a cameo appearance in upcoming film Shazam!, about a teenage boy who transforms into a superhero.
However, the actors manager Dany Garcia suggested those reports may be premature, urging fans to be peaceful, the cape is still in his closet.
Many fans were keen that he should stay in the role.
One called the speculation a ridiculous rumour, with other comments such as hope to God youre staying on as Superman, if not be the next Bond. Ur amazing actor.
One said: I have decided to shift to Marvel because Henry Cavill is the only reason for me to become DC fan
Some, however, had other ideas.
The BBC has asked Warners and Cavills representatives for comment.
Why might Superman need a reboot?
IGN Entertainments senior editor Scott Collura wrote that its insane, if not surprising, that Warner Bros doesnt have a Superman any more.
He said Warner Bros dont know what to do with the Man of Steel.
With Cavill apparently out, and [Batman actor] Ben Affleck rumoured to also be ready to hang up the cape and cowl, Warners and the folks behind the DC films are presented with an opportunity to soft-reboot the series, he added.
DC may be looking for a major change of direction for Superman after the runaway success of Wonder Woman, which smashed box office records last year and will be followed by a much-awaited upcoming sequel.
Cavills Man of Steel did good blockbuster business in 2013 but studio expectations were higher, according to Forbes.
Things didnt improve with 2016s Batman V Superman: Dawn of Justice, which got a mixed critical response and was seen as relentlessly bleak as well as a financial disappointment, according to Deadline Hollywood.
Roll on to 2017s Justice League, powered by a DC Comics superhero team seeking some of the huge popularity of the Avengers ensemble films.
But Justice League, with a cast including Cavill, Affleck and Gal Gadot, was an undeniable failure according to Forbes, because it didnt recoup its costs.
Anna Smith adds: In recent years, DC has found it hard to compete with the might of the Marvel movies the Avengers gang are much sunnier and teen-friendly, while the Superman films have struggled to find the right balance between deliciously dark and downright sombre.
Wonder Woman struck a much more upbeat and exciting tone (and in the hands of a female director, take note!).
Forbes concluded: If youre Warner Bros, you cannot simply ignore the fact your Superman has repeatedly failed to live up to expectations critically, financially, or in sheer popularity.
What could happen next?
Deadline is reporting that this could be a chance to shift DCs focus to Supergirl, as well as continuing to ride the success of Gadot as Wonder Woman, with its upcoming sequel set in 1984 (look out for those flicked hairdos).
Another forthcoming film out of the DC stable will be Aquaman, starring Game of Thrones Jason Momoa as Arthur Curry, who learns hes the heir to the underwater kingdom of Atlantis.
And of course Shazam! is due out next year as well.
These films are brighter in tone than their bleaker Superman/Batman predecessors.
BBC