The HSE is working on the 'absolute premise' that the new strain of coronavirus is already in Ireland.
CEO Paul Reid said they haven't identified cases of the variant to date, but it's "hard to believe it isn't here".
Concerns over the new strain have prompted the Government to ban travel from Britain until the end of the year.
Meanwhile, new coronavirus restrictions will come in place here amid a surge in COVID-19 numbers in recent days.
A total of 970 new cases were reported yesterday.
Mr Reid told Newstalk Breakfast that Ireland is at the highest risk position in relation to COVID-19 since the first phase of the virus - and we're seeing 'exponential growth' every day.
He said: "There is no doubt about it - we're in quite a serious and dangerous position in terms of the transmission of the virus right now.
"Certainly what we have seen is a higher number of contacts... we can see people are meeting each other more, out more, shopping and visiting because the opportunity is there. That certainly is a factor."
He said checking for the new strain is now part of the testing process.
He said: "We haven't identified it to date, but that doesn't indicate it isn't here. We're working on the absolute premise it is here."
Tánaiste Leo Varadkar yesterday warned we could see 2,000 new cases a day by New Year's Eve.
Mr Reid, meanwhile, said we're likely to have around 1,000 cases again today 'if not more'.
He explained: "There's no doubt when you get to a certain level... many elements of a health system come under complete stress and duress.
"It may not just be testing and tracing - it could be our hospitals after Christmas and into January."
He said contact tracing pressure isn't just around the number of cases, but also driven by the number of contacts confirmed cases have had.
While the HSE previously announced more detailed contact tracing to find out the potential source of outbreaks, Mr Reid said the primary focus now is "making contact with contacts".
He said HSE staff are scheduled throughout Christmas for swabbing, testing and tracing.
However, he urged the public to really follow the guidelines to reduce the pressure on the health system over the festive period - which is usually one of the busiest times of the year for hospitals.
Meanwhile, planning is continuing for the start of the vaccination programme in the coming days.
Mr Reid said 75,000 staff and residents of long-term care homes will be first in line, as well as frontline health workers.
While details are still being finalised, the HSE chief said they're now hoped to have received around 300,000 doses of the Pfizer / BioNTech vaccine by the third week of February - with around 40,000 doses being delivered each week.