The Minister for Health, Mr Martin, has said sporting organisations should immediately move to phase out funding from alcohol companies.
"I think they should move now. I am unhappy with the fact that we have major alcohol companies and manufacturers as the major sponsorship of a major national game."
Asked if he thought it was reasonable to expect a full ban within five years, the Minister said he did.
Mr Martin said he favoured the reduction of the "dependency on alcohol which certain sporting organisations have . . . through the State getting involved in funding the current aspects of funding requirements of clubs, not just the capital funding, to create a healthier society".
Speaking on RTE's Morning Ireland, the Minister said he would also like to strengthen existing codes and standards for the advertising of alcohol. He has also said he would not "rule out the introduction of a statutory framework for the control and regulation" of alcohol advertising.
Mr Pat Barry, director of corporate affairs with Guinness Ireland Group, said he believed further restrictions on advertising and sponsorship would not deal with such a complex problem as alcohol abuse.
He said advertising of alcohol was aimed at creating brand awareness - so if people were going to drink they would choose a drink they saw advertised - not to increase the overall amount of drink consumed. Furthermore, the State was now part of a global village and advertising of alcohol would still arrive through some channel, said Mr Barry.
A spokesman for the Gaelic Athletic Association (GAA) said any sponsorship it received from alcohol companies was for adult teams only.
The organisation has received no communication from any Government body about the sponsorship. The spokesman said the organisation had four years of a contract with Guinness left to run.
Mr Martin this week launched a National Alcohol Awareness campaign, before the release next week of figures which will show the State's young people as the heaviest drinkers in Europe.
The Government had gone for the "soft option" in having an awareness campaign, the Irish Medical Organisation's Dr Declan Bradford said.
He said specific measures were required, including the banning of the advertising of alcohol, banning sponsorship of sporting activities by alcohol companies, increasing the price of alcohol and reducing the number of outlets for the sale of alcohol.
The chief executive of the Vintners' Federation of Ireland, Mr Tadhg O'Sullivan, said raising the price of alcohol would only penalise those who enjoyed alcohol socially, as people who want to abuse alcohol would do so regardless of price.