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The party has been in talks with the Lib Dems ahead of next week's full council meeting which is due to appoint a Lord Provost, council leader and committee conveners.
Labour's Scottish leadership has ruled out formal coalitions with other parties, but it is understood the Lib Dems could be offered "non-political" posts like the chair of the licensing board and convener of the regulatory committee, in return for helping them into office.
The May 5 council elections left the SNP as the biggest party with 19 seats, Labour has 13, the Lib Dems 12, Greens 10 and Tories nine. The SNP and Greens have announced they are in formal negotiations about forming a coalition, but their combined numbers fall short of the 32 needed for a majority.
Labour could win power at the City Chambers if its motion to appoint its councillors to the key posts was supported by the Lib Dems and the Tories at the council meeting on Thursday.
One insider said there had been "constructive" discussions with the Lib Dems but stressed there had been no talks with the Tories and they would not be offered any posts.
Before it can be proposed at Thursday's meeting, the plan has to be approved by the Labour group; the local government committee, representing local party members and trade unions, must take a view; and finally the proposal has to be endorsed by Labour’s Scottish Executive Committee.
The insider said: "Across the country, Labour groups are offering non-political positions to other parties."
A Lib Dem source said the party had not received a definitive proposal from Labour, but indicated they were open to consider such an arrangement.
The source said with just 13 councillors, a Labour administration would have a lot of jobs to fill. "If the Labour party wants to concentrate on the administration and the Lib Dems can assist in terms of these non-political roles, I think that's something we'd be keen to look at. It's certainly an option that can be explored.
"I struggle to see us voting for an SNP-Green coalition and abstaining would effectively mean the SNP got in by default.”
The Lib Dems previously ruled out any deals with the SNP, accusing them of not being ready to change their approach.
The source said: “Labour has been much more open to recognising some of the mistakes made. We shut things down with the SNP because it was clear to us they were going to take the same approach as over the last five years while Labour were much more willing to recognise they hadn't got everything right and wanting to do things differently.”
But a deal would not be “confidence and supply” where a smaller party guarantees support for a larger party on key votes. "I think if we assist we will remain as an independent opposition political party. Our view has always been if you’re wanting to lock parties into an agreement it really needs to be a coalition, a full partnership – that is clearly not going to happen here and because of that we would want to continue looking at issues on a case-by-case basis."