Ryanair's former deputy CEO told the HSE there seemed to be "no urgency" in efforts to cut spending and that the Government needed to be told certain savings targets would not be met.
In January, Michael Cawley, a member of the HSE board, put forward 14 ideas he believed would save cash for the health service in 2025.
However, by April, he told the HSE CEO Bernard Gloster that progress had been too slow and that a reduction in spending - including in cutting administrative costs - was "neither achievable nor realistic."
Mr Cawley said he appreciated the health service had made efforts to contain costs but that "uch more remains to be done."
His list of recommendations for suggested savings included technology solutions for home care, reducing the amount of spending some management could authorise, and high IT maintenance charges.
He also raised over-prescribing and excess supply of certain medications, as well as the possibility private companies could use equipment and operating theatres in public hospitals during evenings and weekends.
Mr Cawley said eight of his 14 suggestions had not been addressed and asked for closer monitoring of savings on travel, subsistence, postage costs and training.
He said while attending a recent audit meeting, it also became clear that initiatives from the finance department of the HSE were not going to deliver predicted savings.
"f this attitude and approach don't change, I propose that we inform the department that we are abandoning this objective as it makes no sense to continue with the pretence that we are serious about it," he wrote.
In response, the HSE CEO Bernard Gloster said he thought he was clear about which of the 14 suggestions were "doable for now".
"I thought we had bottomed this out for now,"he wrote, saying he apologised if there was any misunderstanding.
"I have made some of the hardest and immediate decisions made by any measure against my predecessors. I have more to do but I can only go as fast and as safe as the system around me which I think many accept is variable in its management capacity," Mr Gloster said.
On whether to notify the Department of Health that certain savings were unlikely to be met, Mr Gloster said he shared concerns over "achievability".
However, he said 2025 was a "daunting" year for the health services and that some of the changes would be "slow to bite" and yield results.
"I share frustration at [the] time and pace of some change. I have told department and indeed government at last Cabinet Committee that while pay control is established [for the] first time ever and we held ground on recent dispute, that I am concerned about non-pay," Mr Glouster wrote.
"Not sure if that equates to abandoning the measures but again happy for [the] board to reflect a view as it is a significant issue," he added.
In response, Mr Cawley said that a shift in the "commitment, energy, and imagination" was needed from key members of the HSE's senior leadership team.
"I have acknowledged the steps you have already taken in relation to cost reduction but unfortunately even if 100% effective their impact will fall significantly short of the target set and what I believe is achievable," he said.
Asked about the records, a HSE spokeswoman said: "It is the role of board members to analyse and question the work of an executive team, and the HSE executive team welcomes this".
Reporting by Ken Foxe
