The newspaper cites witnesses who said that an argument broke out between two groups of Thai youths at Zoom Bar on Haad Rin Beach.
A member of one of the groups reportedly fired a shot into the crowd as he ran out of the bar, hitting Mr Ashton in his torso.
Police have reportedly arrested a 26-year-old suspect and plan to question him at Noon on Wednesday.
The Thai island of Koh Phangan, where 22-year-old Stephen Ashton is believed to have been killed in crossfire between rival gangs, is a popular party destination for tourists.
It is famous for its monthly Full Moon parties on Haad Rin Beach which draw thousands of tourists, especially at this time of year.
The Foreign Office has confirmed the name of the British tourist who was killed in Thailand as 22-year-old Stephen Ashton from Redhill in Surrey.
A British tourist has been shot dead at a New Year's Eve party in Thailand after being caught in the crossfire between local gangs, according to reports.
The 22-year-old was said to have been killed as he danced with friends at a beach bar on the island of Koh Phangan.
He is thought to have been caught in the crossfire when a fight erupted between two local gangs, it was reported.
A Foreign Office spokesman said: "We are aware of the death of a British national in Thailand on New Year's Eve and we are providing consular assistance to the family."