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Sources pointed out there were several high-profile events coming up imminently where the Capital's civic head would normally have a key role – including a civic reception if Hearts win the Scottish Cup at the weekend.
The full council is due to meet on Thursday for the first time since the May 5 elections and the appointment of a Lord Provost is top of the agenda, followed by appointing a council leader, deputy and committee conveners.
Despite the likely lack of any coalition deal by then, officials said the council meeting would take place and it was up to the politicians to decide how far they went with the business. The meeting could be adjourned for a week without making any appointments.
But one source said: "There is quite a strong appetite to get a decision on Lord Provost at least. There is some concern at the moment that we do not have a civic head who can make statements on behalf of the city, for example on the passing of Sir Angus Grossart over the weekend."
The Lord Provost would also normally attend the Church of Scotland General Assembly, which starts on Saturday, and the Ceremony of the Keys at Holyrood Palace when the Lord High Commissioner, the Queen's representative to the Assembly, arrives to take up residence.
Officials said if there was no Lord Provost it would fall to council chief executive Andrew Kerr to host a civic reception if Hearts beat Rangers to lift the Scottish Cup on Saturday.
Talks on a coalition or other arrangement to run the city have been taking place for the past ten days, but a continuation of the SNP-Labour coalition appears to be off the table and speculation has focused on an SNP-Green partnership or possibly a Labour-Lib Dem-Geen deal. But the Greens said they last met the SNP on Thursday and spoke to Labour the same day. Labour had not got back to them and there were no more talks scheduled with the SNP. Labour sources said they had only held talks recently with the Lib Dems.
Lib Dem group leader Robert Aldridge, the city’s longest-serving councillor with nearly 40 years at the City Chambers, has been mentioned as the most likely candidate to win cross-party support to become Lord Provost.
One councillor said: “He has been in administration, he’s been in opposition, he’s been part of the largest group, he’s been part of the smallest group, so he knows all the different aspects of being a councillor. He’s seen as a very fair and reasonable arbiter of council business. And because we’ve got such a balanced council, he would be a very experienced chair in what could be a bumpy five years.”
Other possible candidates for the role include Labour’s Joan Griffiths, who was previously Depute Lord Provost, and the SNP’s Cathy Fullerton.