Greenock-based Caskie Limited will be responsible for knocking down sections of the housing estate, as well as the former church and school buildings.
The company secured the £400,000 contract through an open tendering process.
Demolition work is expected to start early in the new year.
Councillor Michael McCormick, Inverclyde Council's convener of environment and regeneration, said: "This is another key milestone in the redevelopment of Clune Park that we now have a contractor on board to carry out the first phase of demolition works.
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"It's particularly pleasing that an Inverclyde company will be carrying out the work, underlining the council's commitment to supporting local businesses.
"The council spent over £20 million with local suppliers in the 2023/24 financial year, which is almost a quarter of the entire council spend on contracts for that year.
"Congratulations to Caskie Limited on successfully bidding for the contract."
The first phase of the demolition programme will see the demolition of 138 properties across 15 tenement blocks, as well as the former school and church buildings within the Clune Park estate.
They are being demolished in the interests of safety after dangerous buildings notices were served following extensive investigations by council building standards officers and external surveyors.
It is the first phase of demolition works at Clune Park, which comprises around 430 residential properties across 45 tenement blocks.
Euan Caskie, Director of Caskie Limited, said: "We are very proud to be part of this project, securing work for our Inverclyde-based team and helping to improve the local environment."
The first tranche of flats to be demolished are opposite the former church building and comprise of the western side of Bruce Street and connecting section of Robert Street and both sides of Wallace Street.
