A pharmaceutical company making a new cancer medicine is demanding such a high price for the drug that it would cost the State €64m over a five-year period.
The HSE, which has an additional €7m for new advanced cancer drugs this year, said it could not afford to pay out that amount of money.
It was responding to calls by a number of cancer specialists for the HSE to pay for the drug pembrolizumab which, they say, has made a major difference to patients with advanced diseases.
The group of patients had been given the drug, made by Merck Sharp & Dohme, for free on compassionate grounds over the past year.
However, the company now wants the HSE to start paying.
It is being assessed by a group of experts who advise the HSE on the efficacy of a medicine.
However, they must also consider its value for money based on the price demanded.
The HSE has also been called on to make a second drug, Nivolumab, made by Bristol-Myers Squibb, available for treatment of patients with cancer of the lungs or kidneys.
A similar free-access programme which has allowed a group of patients to use the drug on a trial basis is ending next week.
It must also be assessed along with the price demanded.
The HSE said yesterday that it must operate within its allocated budget for 2016, and within this prioritise the allocation of resources across the entire health system.
The high prices drug companies are still charging for some medicines was identified in reports yesterday.
A Cork cancer patient said he can get a six-month supply of the drug Bicalutamide in Newry for €50 while paying €144 for a month's supply in the Republic.
The HSE said it "engages robustly with manufacturers to achieve the best price and the best value for Irish patients and the taxpayer".
A spokesman said it is widely recognised that Ireland is paying higher prices for medicines than many other European states.
"The single biggest barrier to new drugs approvals in Ireland is the high prices that manufacturers are seeking to charge," he said.
"Lower prices would enable the HSE to afford more innovative drugs for more patients as they become available.
"The HSE will continue to engage with manufacturers in the best interests of patients and the taxpayer.
St Vincent's Hospital cancer specialist Prof John Crown yesterday called on the HSE to approve the drugs, saying they were "game-changers".