NHS Grampian and NHS Ayrshire and Arran both asked for help last week, but the requests remained under consideration by the Scottish Government.
They have now been forwarded to the MoD.
It comes as health boards around the country continue to struggle under mounting pressure and staff shortages.
NHS Lothian warned the public on Tuesday to only come to A&E if their condition is “life-threatening”.
Asked by Conservative MSP Sue Webber on Wednesday if he would endorse this, health secretary Humza Yousaf said it echoed advice from all health boards not to visit A&E unless a patient’s condition is critical or life-threatening.
Mr Yousaf added that members of the public “should not hesitate” if they are concerned that they or a loved one is in a critical or life-threatening condition.
He added: : “Our health and care system is under extreme pressure due to the pandemic, and NHS Lothian, like, of course, all health boards, is experiencing significant pressure including workforce challenges, a high level of delayed discharge, which is inevitably affecting waiting times in A&E.
“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.”
The Scottish Government confirmed on Wednesday that a request for support for NHS Grampian and Ayrshire and Arran has been sent to the MoD.
“The requests for military assistance from NHS Grampian and NHS Ayrshire and Arran have been agreed by the Scottish Government and Military Liaison Officers, and were today submitted to the Joint Military Command in Scotland for their consideration,” a spokesperson said.
A spokesperson for NHS Grampian said they were “optimistic” that the request would be approved, and grateful to the MoD for considering it.
NHS Lanarkshire and NHS Borders are already receiving support from 86 personnel, most of whom are Army medics.
Opposition MSPs called on ministers to approve the requests for help at Holyrood on Tuesday.
When asked by Scottish Conservative leader Douglas Ross 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 Scottish 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.”