Staff working at the Queen Elizabeth University Hospital said the A&E department continued to be overwhelmed.
NHS GGC, which declined the move, said it was taking action to support staff.
It comes as both NHS Borders and NHS Grampian issued a plea to staff on leave to come into work, due to "extreme pressure" at hospitals.
With an increase in flu and respiratory viruses across Scotland, health boards have asked the public only to attend A&E if it is urgent or life threatening.
A series of messages seen by the BBC show a request from A&E consultants in Glasgow to declare a major incident was declined by NHS GGC on Thursday.
They asked for it to be called so extra resources could be diverted to the department.
A major incident is declared when the the health of the community is under serious threat.
Special measures would be put in place to assist the demand on the health service, such as extra resources to manage the volume of patients.
The messages said every space in the emergency department was occupied by patients waiting for a bed in the hospital, with no resuscitation beds and an eight-hour delay for ambulances offloading patients.
Dona Milne, NHS Lothian director of public health, said: "We have now reached extraordinary levels of flu in Scotland that we haven't seen for many years.
"Although they can be mild infections for many, it can be extremely serious for some people causing them to require urgent hospital treatment.
"We are asking anyone who has cold or flu like symptoms to help limit the spread of infection. Stay at home if you feel unwell. And if you do have to go out, please wear a mask."
Scotland recorded its worst ever performance times at A&E in the week up to 18 December, with 55% of patients seen within the government target of 4 hours.
This is down 7.4% compared to the previous week.
At the Queen Elizabeth University Hospital, just 45% of patients were seen within four hours.
A total 1,821 patients spent more than 12 hours in emergency departments across Scotland.