There has been a "relatively stable number" of new monkeypox cases being identified each week, according to the Director of the National Immunisation Office.
Dr Lucy Jessop has said that 97 cases of monkeypox have been identified in Ireland so far this year.
Speaking to RTÉ's News at One, Dr Jessop added that the number of cases is "not increasing at an alarming rate".
The HSE set up an incident management team, which has been meeting since the first cases were identified in May.
"We've been closely monitoring it and looking at all aspects of the identification, care and prevention of monkeypox," she said.
She added that in line with the National Immunisation Advisory Council advice, people who are close contacts of cases of monkeypox are being offered vaccination.
She added that NIAC has recently given additional advice to offer vaccination to some people who are particularly at high risk of contracting monkeypox.
"In terms of people who are particularly at risk of this disease, it's people who have multiple sexual partners, including some groups of men who have sex with men," Dr Jessop added.
"In terms of the risk of the broader population, that risk is very low, so it isn't a vaccine programme that will be offered more widely than that."
The UK Health Security Agency (UKHSA) said tonight there were "early signs" that the monkeypox outbreak is plateauing across the country and that its expansion has slowed.
"While the most recent data suggests the growth of the outbreak has slowed, we cannot be complacent," Dr Meera Chand, director of clinical and emerging infections said.
There were 2,859 confirmed and highly probable cases of monkeypox in the UK as of 4 August with nearly 99% of the cases among men, the country's health authority said in a statement.
Authorities in the UK in June recommended gay and bisexual men at higher risk of exposure to monkeypox be offered a vaccine, as the outbreak of the viral disease had gathered pace.
Campaigners here yesterday called for the planned HSE monkeypox vaccination programme to begin as a matter of urgency.
HIV Ireland Executive Director Stephen O'Hare said that the planned roll-out had not started yet and the Government is currently putting a vaccination plan in place.
The HSE said that since the multi-country monkeypox outbreak began, the HSE has received limited supplies of smallpox vaccine and has been using this to respond to cases of monkeypox.
It said that the vaccine is being offered to close contacts after a risk assessment and also to some healthcare workers who may be at risk of exposure through their work.
At this time, supplies of vaccine in Ireland and in the EU are low and limited, it added.
The United States yesterday declared the monkeypox outbreak a public health emergency, a move expected to free up additional funding and tools to fight the disease.
The World Health Organization has also declared monkeypox a "public health emergency of international concern," its highest alert level.