The athlete, who has not yet been named, was implicated among the 32 positive tests announced by the International Olympic Committee (IOC) last month following re-tests of old samples using new technology.
Jamaican website RJR News reports that the Jamaican Olympic Association (JOA) and the athlete was notified of an A-sample failure last month.
Confirmation will not be given, however, until after the results of the B-samples, which was due to be re-examined here at the World Anti-Doping Agency (WADA accredited laboratory yesterday.
The Jamaican Observer reports that the athlete failed for banned stimulant methylhexanamine.
insidethegames understands that the athlete involved is not Usain Bolt, the six-time Olympic champion and world record holder in the 100 and 200 metres.
Jamaica won six gold, three silver and two bronze medals at Beijing 2008, all in athletics.
It left them fourth overall in the athletics medal table.
Golds were won by Bolt in the 100m and 200m, as well as in the 4x100m relay with Michael Frater, Nesta Carter and Asafa Powell.
Shelly-Ann Fraser was the women's 100m winner and Melanie Walker and Veronica Campbell-Brown won the respective 400m hurdles and 200m titles.
Powell has since served an 18-month ban after failing for supplement oxilofrine in 2014.
Campbell-Brown was cleared of all wrongdoing after initially failed for a banned diuretic in 2013.
Methylhexanamine was the same substance that 400m runner Dominique Blake failed for during the Jamaican Olympic trials for London 2012.
Botswana's 2011 world 400m champion Amantle Montsho and Nigerian sprinter Oludamola Osayomi are among athletes to have tested positive for it.
Three athletes also failed for the substance during the Sochi 2014 Winter Olympics: bobsleigh brakeman and former decathlete William Frullani, German biathlete Evi Sachenbacher-Stehle and Latvian ice hockey forward Vitalijs Pavlovs.
Originally developed in 1944 as a nasal decongestant pharmaceutical drug, the substance was withdrawn from sale in 1983, only to be reintroduced 20 years later and distributed under a variety of different names as an energy-boosting dietary supplement.
Many experts have questioned how safe it is to take and in 2010 the American military issued a recall of all methylhexanamine-containing products from all military exchange stores worldwide.
The JOA did not respond for a comment when contacted by insidethegames.
Fourteen Russians in athletics, weightlifting and rowing and one Spanish athlete have already been named as testing positive after the re-analysis of samples from Beijing 2018.
Twelve countries and six sports are implicated, the IOC said.
The situation is due to be discussed here at the second day of a meeting of the IOC's ruling Executive Board.
Full details are expected within the next few days following re-testing of all B-samples.
As well as being stripped of their medal, the athlete concerned in this case could also face a backdated ban of two years, the maximum sanction for the offence at the time of the WADA rules.