The Minister for Health has said September remains the "aspiration" to have every person in the country vaccinated, despite issues around supply.
Stephen Donnelly said the vaccination programme is "going well", but delays to the supply of the AstraZeneca vaccine could hamper the Government's timeframe.
Mr Donnelly said Ireland has ordered about 14 million doses of various vaccines and, based on those figures, it is likely everyone will be vaccinated by September.
"September is absolutely still the aspiration. Yeah, absolutely. It has to be," he said.
"We can't promise... because it's a projection based partly on vaccinations that haven't even been applied for authorisation, and on delivery schedules that still have to be fully agreed," he told RTÉ's Today With Claire Byrne.
"I think people in Ireland are asking a very reasonable question, we're all being asked to stick with Level 5, it's been a tough, tough year.
"I think people are reasonably saying, 'based on the information you have, based on your conversations with the pharmaceutical companies, based on what the scientists are telling you, are we likely to all be vaccinated in May?' No.
"Is it likely that it will be this September? Yes."
However he appeared to row back on this, saying he cannot say it is "likely" everyone will be vaccinated by September.
"If the vaccines come through that we have advanced purchased and if they come in on schedule, then it is reasonable to think that by September every adult could be vaccinated, but with all of those very serious caveats," he added.
Vaccinations for the over-70s are due to get under way in the next two weeks, he said.
It comes as the Government announced that lockdown is to be extended until March 5, and also introduced tighter travel restrictions.
The new measures include mandatory quarantine at a designated facility for people who arrive in Ireland without a negative PCR test taken in the past 72 hours.
Travellers arriving without a negative test could also face a fine of 2,500 euro or a six-month prison sentence.
Visa-free short-term travel from South Africa and South America is suspended until at least March 5.
All passengers entering the country will be subject to mandatory quarantine, a change from the policy of voluntary self-isolation.
The number of daily cases on Tuesday was 928, dropping below 1,000 for the first time in over a month.
Mr Donnelly said of case numbers: "Now we have to be very careful, but they're moving in the right direction.
"So actually the most important decision yesterday wasn't about foreign travel, it was about extending the measures, and my big ask is that we all continue, because it only works if we do it together, is that we stick with these measures because they are working.
"About two-thirds of all new cases in Ireland are now this UK variant. We're watching this African variant, we're looking at the Brazilian variant.
He also said total spread from foreign travel is very low.
Over the last three weeks, there have been around 667 outbreaks recorded, with only one linked to foreign travel.
"The South African variants, the Brazilian variant, and I'm told by experts in the field that there will inevitably be other variants popping up around the place - they're game changers.
"When we had a UK variant that was 60% to 70% more contagious. While we have good evidence that the vaccines will work well on the UK variants, the jury's still out on the South African variant and the Brazilian variant and exactly how they interact with the vaccines, and we're taking them very seriously."
Co-leader of the Social Democrats Roisin Shortall appealed to the health minister to outline the Government's plans after March 5.
She tweeted: "Minister for Health @DonnellyStephen continues to describe the problem rather than outline solutions.
"How many people will be vaccinated by 5th March and what happens then?"