The Nationalists are expected to put forward a list of nominations for committee conveners when the full council meets on Thursday.
But they are viewing at least some of the appointments as “temporary” while they wait to hear whether a coalition proposal agreed between the SNP and Labour groups at the City Chambers will win the approval of Labour’s Scottish executive committee.
The executive has so far refused to endorse the plan and asked for more information.
And there has been speculation that Labour is reluctant to form any kind of alliance with the SNP when the two parties are fighting each other in the general election and that no approval will be given until after June 8.
But that would mean the Capital was without a council administration for more than five weeks following the local elections on May 4.
SNP group leader Adam McVey has written to Scottish Labour leader and Lothian MSP Kezia Dugdale, asking for a meeting to discuss the situation.
But in the meantime he says the SNP is prepared to step in and form an administration.
He said: “My position has been clear that I think the best way forward is an SNP-Labour coalition. That would give stability and also clarity of direction.
“I believe that would still be in the best interests of the city.
“If Labour get permission to enter into a coalition at some point that would be the best way forward for the city, for the development of transport, education and all the services people expect us to provide, and also for the crisis in the care service we need to address.
“But I’m not willing to wait potentially five weeks to start addressing these problems.
“I’m going to try to be in a position where we can sort these issues out.
“The city needs political leadership and the SNP is in a strong position to provide that.
“Our platform has synergies with the vast majority of the council.”
The SNP is now the biggest party on the council with 19 seats, the Tories have 18, Labour 12, the Greens eight and the Lib Dems six.
Tory proposals for a “pan-unionist” coalition of Conservatives, Labour and Lib Dems were rejected by Labour.
Last week, at the first full council meeting since the elections, former SNP group leader Frank Ross was elected Lord Provost, with the SNP and Greens voting together while Labour abstained, but all other appointments were postponed until this week.
It is understood that while the SNP will propose committee conveners are expected to be decided on Thursday, along with membership of the committees reflecting the strength of each party on the council, the post of Deputy Lord Provost may be left vacant for the time being.
An SNP source said talks were being held with other parties ahead of the meeting in a bid to win support for the formation of a minority administration.
The source said: “We believe Labour nationally will at some point give permission to their group to form a coalition, but it won’t be until after the election.
“The move to appoint SNP committee conveners is expected to be a temporary arrangement which would change when a proper coalition is formed.”
Labour group leader Cammy Day has said he hopes to hear from the party’s executive soon.
And at national level Labour sources insist there is no plan to delay a decision until after June 8. SNP-Labour deals have already been approved in some other areas.
But although the Edinburgh Labour group agreed to the coalition proposal there are a variety of views within the party.
Labour’s Local Campaign Forum, which has representatives from constituency parties across the city, refused to back the plan, questioning what the party or the citizens of Edinburgh would get out of it. It asked the group to think again.
Labour nationally has set strict criteria for any coalition, including opposition to austerity, no compulsory redundancies and a presumption against privatisation of services.
One senior source said: “Some council groups have been back to the executive four times with proposals – Edinburgh’s situation is not unique.
“But I believe we would be better staying out of any coalition. An SNP-Labour administration wouldn’t have a majority anyway. You couldn’t guarantee what was going to happen.”