Almost 200,000 Covid-19 PCR tests have been lab-tested in the last seven days.
HSE Chief Executive Paul Reid has urged those who are symptomatic to try to book a test via the HSE's website.
Speaking on RTÉ's Morning Ireland, Mr Reid said they are prioritising referrals from GPs and those who are symptomatic.
"If I just look at from midnight last night for the last seven days, we've now lab-tested 199,250 tests," Mr Reid said. "For the last seven days, 82% of people who wanted a swab got it in less than 24 hours.
"If you are symptomatic, do go to our site and try to book.
"We are prioritising referrals that have come through GPs and we are prioritising symptomatic people."
Mr Reid said a "significant proportion" of those coming forward for testing now are symptomatic.
He advised people to continue to check their booking website throughout the day, as sometimes appointments are freed up.
"During the course of the day, we do release some appointments that were reserved for GPs and may not be utilised," he said.
"But 82% in the last seven days have received their swabbing within 24 hours, and yes on some occasions it will go into the next day, so I would ask the public to bear with us.
"These are levels that nobody could forecast or see. We have scaled up to meet them, but it's a significant demand."
Mr Reid said they have brought in additional testing centres and more private capacity.
There are more pop-up centres from the National Ambulance Service, and they are also looking at other providers, he said.
"So we're putting in more centres, more workforce, more automation, and support from the Defence Forces," he said.
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