Opposition MSPs called on the First Minister to approve a request made by NHS Grampian last week, while a spokesperson for the health board said they were “optimistic” about support from the Ministry of Defence (MoD).
In a Covid-19 update to MSPs on Tuesday, Ms Sturgeon said hospitals across the country are “at or close to capacity”, while the social care system is also under pressure, but that military assistance should only be requested when “appropriate”.
The First Minister announced a further £482 million funding for NHS boards and other authorities to help tackle Covid-19, including £121m for Test and Protect and £136m for the vaccination programme.
The UK Government had earlier this week confirmed the NHS in England was set to receive an extra £5.9 billion in the Autumn Budget, due to be announced on Wednesday.
It comes as NHS Lothian revealed on Tuesday that it had requested mutual aid from other health boards, amid 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 are all “close to capacity”, NHS Lothian chief executive Calum Campbell said.
The health board has asked members of the public to only visit accident and emergency if their condition is “life-threatening”. Mr Campbell warned the health board was in a “serious situation”.
New figures published by Public Health Scotland on Tuesday also showed A&E waiting times in Scotland were the worst on record for the seventh time in the past three months.
Just 70.9 per cent of patients were seen within the Scottish Government target of four hours in the week to October 17.
NHS Lanarkshire, where soldiers are already providing support, last week became the first health board to announce it had been placed at the highest, “black” level of risk, due to capacity levels.
The Aberdeen Royal Infirmary in NHS Grampian also reached this level last week and the health board is operating in the face of “sustained pressure”.
A spokesperson said on Tuesday the health board was in discussion with the Scottish Government and the MoD, and added: “We remain optimistic that we will receive support from them, as appropriate, and are extremely grateful to the MoD for considering our request."
This request has not yet been officially put forward to the MoD by the Scottish Government.
A spokesperson for the UK Government said: "Organisations requiring military assistance must have their request endorsed by a relevant UK Government minister and approved by defence ministers. The Office of the Secretary of State for Scotland has not received a request from NHS Grampian."
NHS Ayrshire and Arran has also requested military assistance and it is understood this has not been put forward to the MoD either.
When asked by Scottish Conservative leader Douglas Ross on Tuesday why the request from NHS Grampian had not been approved, Ms Sturgeon said: “The request for military support by NHS Grampian will be submitted when it has been refined to an appropriate degree.
"It is important that health boards do not request what they do not need and it is important that requests are properly framed. That work is ongoing.”
Pushed again on the issue by Labour leader Anas Sarwar, Ms Sturgeon added: “It is often the MoD, rightly and understandably, that asks for these requests to be refined.
"It's important that we make use of military assistance where that is appropriate, but that we don't ask the military to do things that health boards should be doing themselves, and that's why that process is so important.”
Mr Sarwar said he was "confused” by the First Minister’s comments on NHS Grampian.
He said: "The simple fact is this. NHS Grampian has asked the Scottish Government for additional support and military assistance, and over a weekly MoD has not received that request.
"We need urgent action for those NHS boards to actually deliver quality NHS care for people.”
Health secretary Humza Yousaf also confirmed to MSPs that no other health board as a whole had yet been declared at the highest or “black” level of risk, as in NHS Lanarkshire.
"Decisions on what procedures and treatments can take place are taken at individual health board level,” he told Labour’s Monica Lennon.
"I speak to health board chairs and chief executives on a very regular basis and all boards are at a significant level of escalation.
"My most recent conversation with chairs was yesterday and there was no indication that any other chair, chief exec or health board was going to declare the highest level of escalation, as Lanarkshire has done, but that is, of course, kept under review.”
In NHS Lothian, hospitals are “close to capacity”, with patients waiting “longer than ever” to be admitted, Mr Campbell said.
He said: “Our hospital system is under extreme duress. We have asked for mutual aid to help ease the sustained pressures our teams and patients are facing.
“We are seeing high volumes of people who are seriously ill and need admitted to hospital right away. On top of that, we have increased levels of general staff sickness and self-isolation, which has placed serious pressures on staffing levels.
“To try and relieve some of the pressure, we are actively re-deploying staff from across the sector into the roles that need them most.
“I cannot praise and thank our staff enough for the ways in which they are continuing to respond to these new challenges. However, there is no denying that we are in a serious situation.”