Labour emerged as the largest group with 20 out of the 58 councillors, while the SNP have 18, the Tories 11, the Greens six and the Liberal Democrats three.
The various groups were holding separate meetings this morning to decide their negotiating positions and face-to-face meetings between negotiating teams for the parties were expected to take place over the course of the weekend.
Councillor Burns said he wanted to form an administration using the “best talents” from all the parties.
But others voiced scepticism about whether such an arrangement could work.
Other options which could create a majority include a Labour-SNP coalition or a Labour-Conservative coalition. A Labour coalition with the Greens would fall four short of a majority, but if the Lib Dems were included it would give them exactly half the seats on the council – the same number which allowed the Lib Dem-SNP coalition to rule for five years with the casting vote of the Lord Provost.
Labour and SNP sources both described a coalition between the two parties as “unlikely”.
Labour played down the idea of a deal with the Greens and Lib Dems, saying “29 is not enough”.
But a Tory source said there was scope for a deal between the Conservatives and Labour, arguing Labour’s policy of a “co-operative council” – giving local people more control of services – offered some common ground. “We want to cut the size of the council and take some stuff out of council control. So that fits well with our ethos, what we want to do,” the source said.
Cllr Burns said he was “ruling nothing out” and signalled that he did not expect a quick result to cross-party talks.
He said: “I’m determined not to be rushed into any hasty arrangement. We’re going to take our time and make sure we have a stable arrangement for the next five years. If that takes a few days, so be it.
“I want to speak to all the parties. My aim is to use the best talents from all the parties. That’s what we said in our manifesto and we meant it.
“Politics as usual has failed in this city over the last five years and we need a radical change.
“I’m not looking for an arrangement where two or three parties are in power and two or three parties are out of power.
I don’t think that’s what the people of Edinburgh have just voted for. That style of politics has had its day.”
He rejected the idea that an all-party administration without anyone in opposition would be undemocratic.
“Backbenchers can be a very effective opposition no matter what party they are from.”
He expressed “cautious optimism” about how other parties would respond. “Some may agree, some may not.”
The SNP, who had hoped to emerge as the biggest party and renew their coalition with the Lib Dems, were non- committal on the grand coalition plan.
Group leader Steve Cardownie said: “We will sit back and wait to see what Labour have to say on the matter.”
He said the party was keeping its options open.
On the idea of a coalition of all the talents, he said: “It will be interesting to see what it means. I’m interested in how that concept can be developed. It’s a novel suggestion. We’ll need to see some meat on the bones.”
However, a senior SNP insider said trying to maintain a coalition of all the parties would be like “herding cats”.
Tory group leader Jeremy Balfour said: “No party has all the answers on all the issues. If we can bring together the best people from within the council that would be a good thing. It is certainly worth exploring. We have always said we would try to play a constructive role and work with anyone to achieve that.”
But a Tory source suggested a Labour-Conservative-Lib Dem coalition could be a better prospect.
“It would be difficult for all the parties to be in coalition together. To get it to work, we would have to exclude the SNP. It has to be a coalition of those committed to the Union. Whether the Greens could be part of that is questionable. It may not be a key issue locally, but for the capital city it is symbolically quite important.
“A Labour-Conservative coalition would have the necessary numbers, but you would probably need a third party to make it credible.”
An SNP-Conservative coalition would have a majority, but a Nationalist source said such a deal was “out of the question”.
He said: “Some councillors would rather resign than do that.”
The Greens said they were ready to talk to any other party. Leader Steve Burgess said the coalition of all the talents idea was “interesting”. He added: “We just need to see what they are proposing.”
Senior Lib Dem Paul Edie said they too were open to discussion. “As a party that believes in proportional representation, we believe in working with people of different persuasions. The elections have thrown up no decisive majority. We would be open to talking to anyone.”
Lib Dem sources suggested an all-party administration would be difficult to hold together, especially in the face of difficult decisions to be made because of spending cuts.