Morgan Sindall, which paid £28m for the majority of Connaught's social housing contracts earlier this month, said it has agreed terms with 42 of the clients concerned.
A statement from the firm said: "We have made good progress with clients, and are pleased to announce that agreements in principle have been reached with 42 local authority and housing association clients for the transfer of their contracts to Lovell.
"Discussions are ongoing with a number of other clients which are yet to agree to the transfer of their contracts."
Stewart Davenport, managing director of Morgan Sindall's social housing subsidiary Lovell, said: "We are pleased to have secured so many partnerships so quickly and are delighted with the way clients have responded. Our prompt action has enabled us to provide customers with essential maintenance services with minimal disruption."