The FA has confirmed it has received a detailed account of issues related to former Cardiff manager Mackay and his head of recruitment Moody, now director of football at Crystal Palace.
It has been reported that the document contains allegations of racist, sexist and homophobic text messages.
An FA spokesman said: "The FA can confirm it is currently investigating this matter."
Mackay is now understood to be out of the running for the vacant manager's job at Palace after the club ended their pursuit of the former Watford boss.
It is understood that the dossier was filed more than a week ago and was not deliberately timed to affect Mackay's candidacy at Selhurst Park.
Former Norwich and Newcastle boss Chris Hughton is understood to have expressed an interest in succeeding Tony Pulis at the south London club.
Palace must now turn to other targets, with former England manager Glenn Hoddle reported to have ruled himself out of a switch from QPR.
Former Tottenham boss Tim Sherwood also looks to have distanced himself from the role, after initially being deemed second choice to Mackay.
Caretaker manager Keith Millen is now expected to take charge of Saturday's home Premier League clash with West Ham.
Crystal Palace declined to comment on the situation regarding both Mackay and current Palace director of football Moody.
Cardiff owner Vincent Tan sacked Mackay in December 2013, just a few months after the Scot had ended the Bluebirds' 51-year wait for top-flight football.
Two months earlier Moody had been placed on gardening leave before being subsequently dismissed.
Mackay launched a £7.5million legal claim against Tan for compensation and wrongful dismissal after his sacking but suddenly dropped the claim in May and issued an apology to the Malaysian businessman.
Cardiff have declined to make any public comment on the dossier, saying it is a now a legal matter in the hands of the Football Association.
Earlier this week Palace were fined by the Premier League for their part in the 'spygate' saga involving Cardiff last April.
The Premier League determined that Palace had breached their 'good faith' rule - B16 - by obtaining information about Cardiff's team ahead of their 3-0 win when the two clubs were relegation rivals.
The Welsh club had complained to the Premier League that Moody had contacted Cardiff employees for information in the build-up to the game.
The Gay Football Supporters' Network (GFSN) admitted shock at the reports of Mackay's alleged homophobic comments.
"The GFSN is saddened to hear reports of alleged homophobic comments by Malky Mackay," said spokesman Simon Smith.
"Homophobia is still a problem in football and we expect a high standard of conduct from managers, who should be setting a positive example of inclusion.
"We will be following the story closely and asking the FA to fully investigate the allegations."
Mackay's representatives were unavailable for comment.