It comes after the health board issued a warning on Tuesday, telling the public to only visit accident and emergency (A&E) if their condition was “life-threatening”.
Chief executive Calum Campbell said services were in a “perfect storm” caused by staffing and bed pressures, combined with high volumes of patients presenting with complex and serious cases.
The Royal Infirmary of Edinburgh, St John’s Hospital, the Western General Hospital and the Royal Hospital for Children and Young People were all close to capacity, Mr Campbell said, and patients were waiting longer than ever before to be admitted.
It has now been confirmed the health board is “actively considering” a request for military aid to help cope with the pressure.
Scottish Labour Health spokesperson Jackie Baillie labelled the move “yet more damning evidence” that health secretary Humza Yousaf has “lost control” of the situation in Scotland’s NHS.
"That yet another of Scotland’s largest health boards is on the brink of drafting in the army to support NHS services is deeply worrying,” she said.
"Across Scotland, NHS services are at breaking point and lives are on the line.
"The health secretary must act now to prevent the situation worsening across Scotland."
NHS Lanarkshire and NHS Borders are already receiving support, while requests from Grampian and Ayrshire and Arran have been put forward to the Ministry of Defence (MoD).
NHS Greater Glasgow and Clyde has also said it is considering asking for help.
Soldiers continue to aid the Scottish Ambulance Service and a request for support with the flu and Covid booster vaccine programmes is under consideration.
A spokesperson for the Scottish Government said on Thursday that no further requests from health boards have yet been made.
Bids for support are first considered by the Scottish Government and military liaison officers.
If approved, they are then sent to the Joint Military Command in Scotland, and then onwards to the MoD.
Asked about NHS Lothian’s warning, Mr Yousaf told MSPs on Wednesday this echoed advice from all health boards.
He said: “We recognise that some people aren’t getting the service neither they nor we would expect and I apologise to anyone who suffers as a result of that.”