The 26-year-old was arrested in the early hours of Monday morning on suspicion of actual bodily harm following an incident outside a nightclub in Bristol.
Police released the all-rounder without charge but the investigation is ongoing and he was absent for England's ODI win over the West Indies on Wednesday.
In the one-minute video, a man resembling Stokes is seen fighting with two men and punches one of them clean out onto the street. One onlooker can be heard saying: "That's enough Stokesy."
The newspaper speculates that Stokes could now be stripped of his vice-captaincy and even be axed from the upcoming Ashes series in Australia, despite being named in the squad on Wednesday.
In a statement, the ECB said: "We have seen this footage for the first time tonight - when posted by The Sun. There is an ongoing police investigation, which will look at all available evidence, and we do have to respect that process."
Stokes suffered a broken hand as a result of the scrap.