Taoiseach Enda Kenny did not ask HSE chief executive Cathal Magee to reconsider his decision to quit his post during a meeting between the two.
The meeting took place as the Health Service Executive issued warnings to the nation's public hospitals they must immediately begin axing all extra spending.
Mr Magee emerged from a meeting with Mr Kenny and declared he was sticking with his decision to leave his post.
He said the one-hour meeting was a "courtesy" and he outlined his reasons for leaving the top job in the health service. He revealed Mr Kenny did not ask him to change his mind.
The new role of director general of the health services being introduced by the Government and Health Minister James Reilly was "very different" to the role of chief executive for which he had signed up. The HSE itself is also being replaced.
He later admitted there were tensions in his relationship with Minister Reilly but stated that they were not the reason for his decision to leave. He said he had a very business-like relationship with the health minister and their exchanges at times were very robust and very frank.
He said health policy was a matter for the minister, not the chief executive.
Mr Magee said his decision to forego a €1m compensation package was because such a payment to him would have been entirely inappropriate in the current economic environment. Such funds would have had be taken from the health service's budget, he said.
Meanwhile, the HSE has sent a directive to the country's public hospitals to warn immediate cuts in health spending must be implemented, but in such a way as to minimise risks to patient safety. The HSE declared it wants a reduction in the average length of stay for patients, more savings on medicine charges, and more cuts in overtime.
The directive stated overtime should only be approved as an exceptional measure and all administrative or managerial overtime must cease at once.