''It was never meant to be heard on-air. I am extremely sorry if I offended anyone. It was the last thing I wanted to do,'' Jones is quoted to have said in his apology to Amla by local media here.
The Aussie expressed his regret to South Africa's media manager Gordon Templeton.
''I believe everyone should get away from the myth that Muslims with beards are suspect. I am friends with everyone in the Pakistan team and have nothing but respect for the Muslim faith. I am truly sorry that I said such a dumb thing,'' he added.
However, having already lost his job after broadcasters of the Test series TEN Sports sacked him, Jones now faces the wrath of Cricket South Africa (CSA) which is planning to report the matter to ICC.
''We take the strongest exception to this comment. The switchboards of both Cricket South Africa and SuperSport, that takes a feed of the broadcast to South African audiences, have been jammed with calls from some very angry people,'' an angry CSA CEO Gerald Majola said in a statement.
''This kind of insulting racial stereotyping has no place in cricket and must be stamped on swiftly. The ICC has strongly condemned racism and we will be discussing the matter with them,'' he added.
Jones made the controversial remark while commentating in the ongoing second Test between the Proteas and Sri Lanka. After Amla had grabbed a catch in the match, Jones had said ''the terrorist has got another wicket''.
Recalling the racist taunts directed at the South Africans during their tour Down Under, Majola said strong action should be taken against such behaviour.
''We just cannot get away from the Aussies as far as such racists incidents are concerned. Racist remarks such as this one by Jones have no place in cricket and action must be taken immediately against this,'' he said.
Meanwhile, TEN Sports has also apologised for the incident, saying that it was willing to work for the development of cricket in South Africa. The channel's CEO Zahib Moorani contacted Majola and an official apology for the incident will be broadcast today.
''He told me that, as part of TEN Sports' apology for this awful incident, he was willing to become involved in social cricket development in South Africa. Exactly what it will involve will have to be determined,'' Majola said about his discussion with Moorani.