It is understood that no FA charges have been brought against the 61-year-old and he has not been suspended.
Moore did not answer calls on Friday while Tranmere released a brief statement which read: "Tranmere Rovers Football Club are aware of media reports regarding Ronnie Moore. As this is an on-going matter the club will make no further comment at this time."
The FA would neither confirm nor deny that Moore was the subject of an investigation by its integrity unit, which deals with betting matters.
Moore became manager of Tranmere for a second time in 2012 following spells at Oldham, Southport and Rotherham.
The Merseyside club are two places above the Sky Bet League 1 relegation zone with 16 matches of the season left.
Already in the last 12 months Accrington Stanley managing director Robert Heys failed in his appeal against the length of a 21-month ban from football for breaking betting rules.
In August, Heys admitted committing 735 breaches of the Football Association's rules over 10 years and was also fined £1,000.
Crystal Palace player, Cameron Jerome, then at Stoke was fined £50,000 for breaking Football Association betting rules back in August.
The striker, 27, admitted repeatedly breaching the regulations, but the charge was not related to match fixing or betting on games he was involved in and he didn't receive a ban.
Three players from the non-league side AFC Hayes did receive bans though after admitting multiple breaches of betting regulations.
Lawrence Shennan was suspended for two years, Ben Goode for one year and Chevy Hart, who is no longer with the Southern Football League club, was banned for four months by a Football Association independent regulatory commission.
And, in Scotland only last month Michael Moffat a striker with League One club Ayr United, was banned for six games for breaking the rules.
The 29-year-old was found guilty of placing seven bets on six matches involving his own team as well as an additional 150 bets on other games.
Last year, Glasgow Rangers midfielder Ian Black was fined £7,500 and banned for 10 matches - of which seven were suspended - after admitting to betting on 160 matches over a seven-year period, including against his own team on three occasions.