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Talks involving various parties have been taking place since the result of council elections became known on May 6, but sources do not believe any agreement is imminent.
The Labour group executive is understood to be due to meet today and one source said any decision made there could be key to the eventual outcome.
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The SNP is the biggest party on the council with 19 seats, Labour has 13, the Lib Dems 12, Greens 10 and Tories nine.
An SNP-Green pact was seen as the most likely scenario since Labour nationally had ruled out formal coalitions. But SNP group leader Adam McVey made clear he would prefer a renewed partnership with Labour because it would have a majority, which a coalition with the Greens would not.
Labour group leader Cammy Day is understood to want a continuation of the SNP-Labour relationship too, but he has not managed to persuade either the national party or the Labour group.
A possible "traffic light" coalition of Labour, Lib Dems and Greens has been floated but there is scepticism about how enthusiastic any of the parties might be.
A source said: "People are hoping we will get some clarity from today's Labour meeting over what they are going to do or there is a strong chance that Thursday's meting may need to be postponed. As the second biggest party, it sits with them to say what they are doing.
“Without Labour deciding whether they are prepared to do something about an alternative administration of some kind it is difficult to see how there would be any alternative to an SNP-Green arrangement.”
The Greens, however, have said a full coalition is not necessarily their preferred option. They are also considering a looser co-operation or a “confidence and supply” deal, where a smaller party guarantees to support a larger party on key issues to keep them in power.
Thursday’s meeting, the first since the election, is due to appoint a Lord Provost and then move on to vote for a council leader and all the committee conveners. If no agreement has been reached on a new administration before then, the meeting could still seek to vote on the Lord Provost and put off the other appointments or adjourn the whole meeting for a week.