The commission instigated talks after the council spent £155,000 suing Kids in Communication, a small voluntary organisation in Wolverhampton, for breach of a contract worth £119,000.
But Seb Elsworth, head of policy at Acevo, accused the council of failing to take these talks seriously by not sending Mark Haysom, its chief executive. Elsworth said: "We felt there was not a great deal of clarity about what the meeting was trying to achieve."
Stephen Bubb, chief executive of Acevo and an NCVO representative, will meet the LSC on Tuesday.
The NCVO declined an invitation to attend the Compact Commission's talks because it wants to discuss the Kids in Communication case first. Its Compact Advocacy Programme lobbied on behalf of the organisation and branded the LSC "pernicious" and "bullying".
"We want to focus on this specific case," said a spokeswoman for the NCVO.
Rob Smith, chief officer at Kids in Communication, is demanding the LSC repay the £50,000 his co-director at the time paid to settle the matter out of court. Smith's MP, Sir Patrick Cormack, has written to the Compact Commission expressing his concern over the skills council's actions.
But the Compact Commission wants to focus on the lessons learned and is planning further talks with the Department for Education and Skills and the Office of the Third Sector.
Rob Wye, director of strategy and communications at the council, said it wanted to create better links with the voluntary sector. "It is difficult to deal with the third sector because it is a broad church," he added.