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But the two parties together are still unlikely to have the 32 seats they would need for an overall majority, so might also seek a less formal deal with one of the other parties to help them into power and support them on key issues.
The count is taking place at the Edinburgh International Conference Centre and results for each of the city's 17 wards will be announced at intervals throughout the day, with the full picture expected to be known by late afternoon.
The SNP is confident of retaining its position as the biggest party on the council and was hoping to add one or two to the 19 seats it won last time. But a turnout said to be well down on the last council elections in 2017 makes the results harder to predict.
The Tories, who won the biggest share of the vote in 2017, were expected to fall back, mainly due to the unpopularity of Boris Johnson and anger about Partygate. One source suggested the party could be reduced from 18 councillors to 15 or 16. Labour's performance was forecast to be much the same as last time, when they won 12 seats.
The Lib Dems, who currently have six seats, were optimistic about getting extra councillors elected in at least two of their target wards, Drum Brae/Gyle and Corstorphine/Murrayfield, and possibly also Almond where they are aiming for a third seat in the four-member ward. The Greens had high hopes of boosting their numbers too, particularly in Inverleith.
A renewal of the existing SNP-Labour coalition seems unlikely in view of Scottish Labour leader Anas Sarwar's stance against the party being involved in formal coalitions. He has argued instead for parties to work together on an issue-by-issue basis.
But the Greens have frequently supported the current administration in crucial votes and the party is in partnership with the SNP at Scottish Government level, making them the natural choice for a deal.
An SNP insider said: "The Greens and ourselves have already agreed effectively that it's the obvious option to go for. But it will depend on how demanding the Greens think they are entitled to be."
It is understood the SNP has a negotiating team, including group leader Adam McVey, housing convener Kate Campbell and transport convener Lesley Macinnes, ready to engage in talks.
However, another source said the Greens were reluctant to go into a coalition if it did not add up to a majority because it would risk defeat on every vote. The new partnership might therefore seek support from another party. Given the SNP's ban on deals with the Tories they would turn to the Lib Dems or Labour.
Some believe that despite Anas Sarwar's opposition to coalitions, the Labour leadership in Edinburgh would be ready to agree some deal short of a coalition, which could still involve at least some posts in the administration.
But there is likely to be opposition among local Labour activists to anything that ties the party once again to the SNP. And some of the candidates expected to be elected are also understood to be against too close an association with the SNP.
Labour group leader Cammy Day could be challenged for his position when the newly-elected council group meets for the first time after the results are known.
There is also speculation about who will become Lord Provost. SNP councillors Cathy Fullerton, Denis Dixon and Norrie Work are all said to be interested in the role. But the post may have to be offered to another party as part of a deal to secure their support. Greens Steve Burgess and Susan Rae have been mentioned. Or if a minority SNP-Green coalition is looking to entice Labour or the Lib Dems into an informal agreement, it could go to Labour's Joan Griffiths or Lib Dem leader Robert Aldridge, who is the longest-serving member of the council.