Northern Ireland's six health trusts say the number of people with Covid-19 in hospital will double by the third week of January.
The warning came in a joint statement released by the trust chief executives on Sunday.
At one stage, a major incident was close to being declared at one hospital, and there were appeals for staff to come into work.
They said the surge in expected hospital admissions cannot be solved by creating more beds because the staff are not there to care for increased numbers of patients.
The Western Trust issued an appeal to staff at South West Acute Hospital in Enniskillen in County Fermanagh.
The appeal said: "Attention all off-duty staff in the vicinity of SWAH. Due to increasing pressures this evening on the NI Healthcare System, we are appealing to you to contact or go directly to the hospital. Thank You," it said.
The trust later added that it had been "overwhelmed by offers of help".
It said: "Thankfully the situation has since stabilised and whilst the divert is still active, we are no longer anticipating patients being diverted to the South West Acute Hospital this evening."
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Northern Ireland's health system has come under severe pressure since Christmas , with hospitals almost at full capacity.
The joint statement from the trusts said: "The situation is very serious with modelling projections indicating that in the third week in January we will be trying to contend with double the number of Covid positive patients compared to the current position today, when several hospitals already have record numbers of patients.
"This is not a simple matter of putting up more beds. We need the staff to care for the increased number of patients.
"Pre-existing staffing pressures and staff absence because of Covid, and other reasons, mean that those staff simply aren't there."
The health chiefs said several trusts had already had to stand down all but the most urgent elective surgery, including some red-flag cancer surgery, to redeploy staff to meet the urgent and immediate needs of extremely ill patients needing ICU care.
"These postponed operations will be rescheduled as soon as possible," they said.
"We have established a regional approach to ensure that any available theatre capacity across Northern Ireland is allocated for those patients most in need of surgery, both during surge and as we come out of this surge.
"This may mean that patients will need to travel further for their surgery.
"Cancer services are seeking to maintain chemotherapy, radiotherapy and other non-surgical treatments and alternative treatments will be provided in the absence of surgical options."
They added that healthcare staff would do everything they could do to deal with the unfolding situation, despite being "exhausted".
"It will definitely not be easy and the care that we are able to provide will at times fall short of the high standards we normally deliver but we will do our very best," they said.
"Desperately ill patients whether Covid or non-Covid will always be the ones being prioritised."
The health trusts urged members of the public not to attend emergency departments in Northern Ireland at any time unless they need emergency care.
They urged the public to continue to play their part by staying at home, practising social distancing and wearing face coverings.
"Never has the phrase 'all in it together' been so pertinent and just so important," they added.
"The Covid-19 vaccines provide the long-term hope and the current lockdown offers the opportunity to shorten the duration of the current surge.
New lockdown restrictions preventing people from leaving home except for essential reasons came into effect on Friday.
The strict rules will remain in place until 6 February but will be reviewed later this month.