A&E departments across the health board are all operating well over capacity, officials said, which is having a “huge impact” on service provision.
NHS Lothian said its services were now working under “the most extreme and sustained pressure”, and the health board has been forced to prioritise capacity on the most urgent patients.
It comes as health secretary Humza Yousaf said the NHS faces “really difficult decisions” in the coming week as pressure is expected to mount.
And Scotland recorded its one millionth case of Covid-19 since the start of the pandemic.
Dr Scott Davidson, deputy medical director for acute services at NHS Greater Glasgow and Clyde (NHSGGC), said staff were under “severe” pressure.
“Our A&Es remain open and continue to assess, treat and admit emergency patients,” he said.
"However, they are extremely busy and staff are facing huge challenges to ensure we’re able to treat patients safely and as quickly as possible.
"We continue to see large numbers of people attending with symptoms that could be managed by speaking to a GP, local pharmacy or by calling NHS24 on 111 before attending A&E.
“As a result of current demand at A&E, people are facing long wait times and our staff are under severe pressure.
“There are other services geared up and ready to see and treat you. If you’ve had a slip, trip or fall, we have numerous Minor Injuries Units (MIUs) across NHS GGC which can look after you, including if you’ve broken a limb.”
Jacquie Campbell, chief officer of acute services at NHS Lothian, said the health board had also seen staff absences increase.
"Winter is traditionally the busiest time of year for the NHS, but Covid-19 has meant that our services are now working under the most extreme and sustained pressure due to a combination of staffing and bed pressures, combined with high volumes of patients presenting with complex and serious cases,” she said.
“We continue to focus our capacity on the most clinically urgent patients and have a process to prioritise those most in need.”
Ms Campbell urged members of the public to consider calling 111 or visiting a minor injuries unit or pharmacy if appropriate.
"The number of staff absent from work can increase in line with rising numbers of positive cases across Lothian and we would ask the public to be vigilant and continue to take the necessary steps to reduce the spread of Omicron,” she said.
Willie Duffy, head of health at Unison, said the union’s members in health and social care were increasingly under strain.
“We have repeatedly been contacted by our members to say it's very, very difficult,” he said.
“There are a lot of staff absences through isolation and various other things. It’s basically non-stop for them, there is no respite.”
Scottish Labour’s health spokesperson Jackie Baillie labelled the call from NHSGGC “incredibly worrying”.
"Everything possible must be done to support NHSGGC staff as they continue to work under immense pressure,” she said.
“The fact of the matter is that the SNP’s complete failure to support A&E departments, have adequate numbers of health and social care staff, and deal with delayed discharge is now putting lives at risk.
"There are significant reported delays in getting through to NHS24, with people waiting over an hour for calls to be answered.
"This must also be urgently addressed by the SNP government to ensure that people are able to access appropriate care quickly.
“Humza Yousaf must act to bring this situation under control immediately.”
Scottish Conservative MSP Dr Sandesh Gulhane said the call was “damning evidence” the NHS was unprepared for the pressures of winter.
“A&E services have been overwhelmed for months on end and now patients are being told to try and access treatment elsewhere,” he said.
“While patients should be directed towards the right care, they will not be reassured by this messaging from the country’s largest health board.
“Humza Yousaf must guarantee no patient will be turned away from A&E if they do require urgent care and get on top of the crisis that is continuing to engulf all areas of our health service right now.”
Scottish Liberal Democrat leader Alex Cole-Hamilton said the news was “alarming”.
“NHS staff continue to be under extreme pressure and patients are paying the price,” he said.
Mr Cole-Hamilton added: “After almost two years of this pandemic and the new Omicron variant, the Government should have had a real plan in motion.
"Patients and staff deserve new hope and the assurance that, in the unlucky case they need it, their local A&E will not turn them away.”
Humza Yousaf told the BBC’s Good Morning Scotland on Thursday the NHS was facing pressure “on a number of fronts”.
“I don’t think there’s any way of sugar coating the fact that these next few weeks will probably be amongst the most difficult our NHS has ever faced in its 73-year existence,” he said.
“We are facing pressures on a number of fronts. We have the pandemic backlog that has built up over the last 21-odd months, we have high levels of staff absences, with increases because of Omicron and its higher transmissibility.
"Then, of course, we are also treating people, over 1,200 people who are in hospital with Covid.”
He added: “This is going to be an extremely challenging period.”