The HSE is "not in a position" to confirm if Ireland will reach its target of vaccinating 80% of adults by June, the Health Committee has heard.
The country also does not have the vaccine supply to hit its target of administering 250,000 jabs a week.
The Health Minister and public health chiefs had hoped to hit the target by April but at present between 140,000 and 160,000 jabs are being administered weekly.
The committee heard there have been 12 changes to either dates of delivery or volumes in the past week, including changes to the AstraZeneca supply in the last 48 hours which will "reduce capacity" over the coming weeks.
Damien McCallion, the HSE national director responsible for the vaccine rollout, said he was "not in a position at the moment" to confirm if the June target would be achieved.
This is primarily because of issues with supply, he said, adding: "There's a very immediate effect of that into the next couple of weeks."
He said the capacity to reach 250,000 jabs a week is in place through hospitals, vaccination centres and pharmacists, but supply levels are preventing that from happening.
He told the committee on Tuesday: "We will get to those figures. We will need to get to those figures.
"But I'm just saying that at the moment, our constraint really is we do not have the supply that would allow us to test the system we were hoping to get there.
Mr McCallion said that for the month of April, the HSE is expecting to receive 800,000 vaccines.
Around 1.4 million are expected in May, and 1.6 million in June.
The figures do not include the Johnson & Johnson single-dose vaccine, of which Ireland is to receive more than 600,000 doses by the end of June.
The J&J jab was reviewed by the European Medicines Agency on Tuesday after fears over rare instances of blood clotting saw the manufacturer pause its supplies to Europe.
The EMA found that the benefits of the jab outweigh the risks.
Professor Karina Butler, chairwoman of the National Immunisation Advisory Committee, said the EMA's findings will be reviewed over the coming days.
Earlier a European commissioner said Ireland will receive enough vaccines to fully inoculate 70% of the adult population by mid-July.
Internal market commissioner Thierry Breton said the EU is on target to supply Ireland with almost six million doses by July.