Both the Taoiseach and the HSE CEO have previously said a ransom will not be paid.
The perpetrators of the cyber attack against the HSE have threatened to publish and sell stolen patient data from Monday, 24 May, according to Bloomberg.
The international news agency reports that the threat was made in order to pressure the Government agency to pay the hackers' ransom of nearly $20 million.
Last Friday, the HSE announced it had temporarily shut down its IT system after it was targeted in a "significant ransomware attack".
Minister of State for eGovernment Ossian Smyth has described the attack as possibly the most significant cybercrime attack on the Irish state.
It has caused major disruptions to health services across the country and it could be weeks before they return to normal.
Both Taoiseach Micheál Martin and HSE CEO Paul Reid have said a ransom will not be paid to the perpetrators of the attack.
An online message, which was reviewed by Bloomberg News, was sent from the hackers to the HSE on Wednesday.
In it, the hackers said that if an agreement could not be reached soon, that they would "start to sell and publish" the HSE's data on 24 May.
The Financial Times reported on Wednesday that 27 stolen files were offered online by hackers, calling themselves the 'ContiLocker Team', as "samples" to prove they had confidential information and to further their ransom demands.
The data included 12 Irish patients' medical and personal information, as well internal health service files, such as the minutes of meetings, equipment purchase details and correspondence with patients.
Conti is the type of cyber attack perpetrated on the HSE and is associated with Russian hackers known as
Wizard Spider.
The Financial Times says the data was offered in a chat between the ContiLocker Team and an unnamed user, which can be viewed on the internet and dark web.
In regards to a ransom,
the Taoiseach said previously: "We're very clear. We will not be paying any ransom or engaging in any of that sort of stuff."
Reid also said: "No State wants to leave its national infrastructure exposed to ransoms and that's exactly what these organisations set out to do."
