Aberdeen, Glasgow, Renfrewshire and South Lanarkshire split from umbrella group Cosla, which still represents the other 28 councils, in 2015.
Cosla remains the only forum for official government funding negotiations, so the breakaway Scottish Local Government Partnership lodged papers at the Court of Session on Friday to take legal action to assert their right to be formally consulted.
Ms Constance has now written to the SLGP offering a meeting - but stressed the government “will only negotiate with one body and that is Cosla”.
SLGP hailed the offer as a “huge breakthrough” in their relations with the Scottish Government.
SLGP convener Jenny Laing said: “They were never going to give us equal negotiating rights with Cosla straight away but the fact that they have both recognised their statutory duty to talk to all local authorities, combined with the offer of bilaterals with ministers and the SLGP, clearly shows that a process is now under way and that will be welcomed by the 1.3 million people we represent across Scotland.”
She added: “The members in the SLGP represent 47% of Scotland’s economy. We have the ideas, passion and drive to grow Scotland’s wealth. All we need now are the powers to do it.”
She continued: “The reason we left Cosla was because we felt powerless to fight the cuts being handed down.
“Nicola Sturgeon is beginning to see sense, which is good news for everyone. The political landscape is shifting rapidly and we all must work closer together in a new era of collaboration in order to get the best possible deal for ordinary, hard-working families.”