Billy Burke, aged 29, who has cystic fibrosis, appealed for Mr Martin to intervene earlier this week, but Mr Martin told the Dáil any decision on a transplant must be a clinical judgment.
A Manchester hospital has offered to carry out the transplant, but the Freeman Hospital in Newcastle, which has "first call" on donated Irish lungs, will not release a lung for him.
After the Irish Examiner highlighted the case, Mr Martin told the Dáil earlier this week everything possible was being done to provide Mr Burke with a lung transplant. He insisted patient selection must continue to be a matter of clinical judgment, whether the patient receives a transplant in Manchester, the Mater Hospital or elsewhere.
Tomorrow people in Killarney will be asked to sign a petition urging Mr Martin to intervene. On Monday week businesses in Killorglin will close for an hour in a show of support for Mr Burke. The Bishop of Kerry, Bill Murphy, is also being invited to say the Rosary on the day.Mr Burke has been waiting for a transplant for three-and-a-half years. He is confined to his home and on oxygen 24 hours a day.
Earlier this week he said: "I can't believe that after going through so much, after all the medical hurdles I've got over, that something as trivial as this could stop me, that I could be left to die because lungs can be sent to one hospital but not to another.
"If it was some medical issue I could accept it, but to find out that it's the result of a political agreement and that I might die because of that I find it very hard to accept."
The Cystic Fibrosis Association of Ireland warned Billy's case is not an isolated one and have also called on Mr Martin to intervene.
The minister said in the Dáil Newcastle did not have exclusive rights to lungs donated in Ireland but has a first call based on suitability. Billy's sister, Lisa, said there was no difference between "exclusive" and "first call".