Former Cardiff manager Mackay has apologised for sending three messages to the club's ex-head of recruitment Iain Moody which he admits were "unacceptable" and "inappropriate"
The Daily Mail published details of four further messages it alleges were of an offensive nature on Saturday.
One of those, it is claimed by the newspaper, makes a racist reference to Cardiff's Malaysian owner Vincent Tan
The other three are of a sexist and homophobic nature, it is alleged.
The FA has confirmed it is investigating the dossier of messages involving Moody and Mackay sent to it by Cardiff, but it may be unable to sanction either man if the messages are deemed to be private correspondence.
Triesman, who served as FA chairman between 2008 and 2010, believes Mackay should be barred from the game for a season solely on the strength of the three messages he admits sending, should the FA have jurisdiction to act.
"There could very well be a significant period in which he can take no part in football," Triesman told BBC Radio 5 Live's Sportsweek programme.
"If it turns out to be three texts then maybe a season (ban), if it's more than that it may be much more."
He added: "Assuming that the evidence is stacked up - I can't pre-try it - I think the FA, in the same sense that it can say to players 'here is a period in which you can take no part in playing', could do so with other people in football."
In May, the FA said it would not sanction Premier League chief executive Richard Scudamore over emails of a sexist nature which he sent.
Current FA chairman Greg Dyke said at the time: "The FA does not as a matter of policy consider private communications sent with a legitimate expectation of privacy to amount to professional misconduct.
"The FA has applied this policy on an ongoing basis and in relation to numerous other cases."
Lord Ouseley, the chairman of anti-discrimination campaign group Kick It Out, believes the case highlights a greater need to address prejudice and bigotry in sport.
He told Sportsweek: "It's very difficult to prescribe what is appropriate.
"My concern is that we're talking about one individual and focusing on that individual and I understand why that is, but we're talking about the wider implications for the game.
"Malky Mackay is probably thinking 'Hey, I'm unlucky, I've got into a situation where all my texts have been forensically examined'
If that happened to every other football manager, senior administrator and other people at top levels of the game then there wouldn't be many people working in the football, the game might just collapse.
"I think what we have got to recognise is that there is a serious problem about prejudice and bigotry and where that is in the hands of people with power, something's got to be done and it's got to be done now.
"The Malky Mackay case is a microcosm that enables us to focus on what is a real, serious problem
If football is going to get to grips with this it has an opportunity now and that's why the role of the FA is important.
"You can say a season, two seasons, whatever, if we think that's it, close it off, we've dealt with that and the problem is over, it isn't."
Source : PA
Source: PA