It is understood the club initially rebuffed the claim, but decided to make a payment when the victim threatened to make the allegations public. It is understood the club did not accept liability.
Heath, who had died before the allegation was made and was therefore unable to defend himself, was Chelsea's chief scout from 1968 to 1979 and discovered some of the club's greatest players, including Ray Wilkins.
A source has told The Telegraph: "He [the alleged victim] thought there were more [victims] but it was never publicised and the case settled. These were very serious offences."
In an official statement published on Chelsea's website on Tuesday night, the club said: "Chelsea Football Club has retained an external law firm to carry out an investigation concerning an individual employed by the club in the 1970s, who is now deceased. The club has also contacted the FA to ensure that all possible assistance is provided as part of their wider investigation. This will include providing the FA with any relevant information arising out of the club's investigation.
"While the club's investigation is ongoing, we will make no further comment on this matter."
The Telegraph is not aware of any evidence corroborating the claim made against Heath by his accuser, although rumours of the chief scout's alleged behaviour had been circulating among some of Chelsea's former players.
However, last night Wilkins, the former Chelsea and England captain, said he was astonished by the accusation. "I knew Eddie very well. It makes no sense whatsoever to me. Eddie was a great guy," said Wilkins. "Eddie Heath was fantastic as far as I am concerned. I have never heard anything like this [about Eddie]."
Heath was an important figure in the Chelsea set-up, at a time when the club was short of money and reliant on a supply of first-team players from its youth set-up. "We were struggling financially and we had to get some home-grown kids through," said Wilkins.
In a match programme in 2011, Chelsea described Heath as an "unsung hero" in a piece on their backroom staff from the 1970s. The programme described Heath as "chief scout who oversaw the recruitment of untold youthful talent for Chelsea between 1968 and 1979, including the likes of Gary Locke, Ray Wilkins, Steve Wicks, John Bumstead, Tommy Langley".
The spotlight has been turned on football following recent claims made by Andy Woodward, a former player, that he was abused as a youth.
The crisis in the national game widened further, with three more police forces - Police Scotland, Staffordshire Police and Cambridgeshire Police - all announcing they had launched investigations into historic sex abuse in football. In total eight forces have begun inquiries, including the Metropolitan Police, Britain's biggest force.
The Football Association has begun its own inquiry, and appointed Kate Gallafent QC to head it up.
On Tuesday former England international Paul Stewart met with Greater Manchester Police at his home over claims he was abused as an 11-year-old.
Stewart, who has waived his anonymity, gave officers the name of his abuser.
Stewart said: "My abuser has not been named yet, and that has been done for legal reasons.
"Today I have spoken to the police and they have opened a formal investigation surrounding this abuser, as a result of my interview this morning."
Tracey Crouch, the sports minister, has contacted the governing bodies of 40 sports amid concerns that historic child sex abuse is not a problem exclusively for football.