The Scottish Local Government Partnership (SLGP) comprises of Aberdeen, Glasgow, Renfrewshire and South Lanarkshire councils.
They broke away from umbrella group Cosla last year.
Ms Constance said the government "will only negotiate with one body".
Cosla remains the only forum for official government funding negotiations, so the breakaway SLGP 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.
The group has hailed the move as a "huge breakthrough" in their relations with the Scottish government.
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."
"The reason we left Cosla was because we felt powerless to fight the cuts being handed down.
"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."
Responding last week to the SLGP's threat of legal action, a Scottish government spokesman said: "We will only negotiate on the final settlement with Cosla, the representative body for Scotland's local authorities.
"However, we have a statutory responsibility to consult all 32 councils on the terms of the local government finance settlement."
Communities Secretary Angela Constance is to meet leaders of a Labour-led council splinter group to hear their concerns about being "frozen out" of funding talks.