Customers at a bar in Fuerteventura, in the Canary Islands, are said to have recognised Mr Edwards.
He was last seen alive in February 1995.
Police believed Mr Edwards committed suicide after finding his abandoned car near the Severn Bridge in south Wales, two weeks after he disappeared. His body was never found.
British barmaid Tracey Jones, who works in the Underground Pub in Corralejo, was reported to have said the drinker was "just like Richey" from the Manic Street Preachers.
"He just started to run towards the door and within seconds he had gone," she told Tenerife's Island Sun newspaper.
The newspaper has now set up a hotline number for anyone who thinks they may have seen Mr Edwards on the island.
Previous 'appearances'
It is not the first time that there has been a reported sighting of Mr Edwards, who would now be 30.
In March last year, a college lecturer said he had seen the missing singer at a market in the Indian resort of Goa.
A spokeswoman for Scotland Yard that the police had not been informed of the latest sighting, but any new information was welcome.
"Our inquiry is active and ongoing," she said.
Fellow band members have always refused to accept he is dead.
One of Britain's most successful rock bands, they have kept aside a quarter share of the millions earned by the group since Mr Edwards' disappearance, hoping that he returns.
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