Crane erectors David Newall, 36, and Rhys Barker, 18, died on 21 June at a building site in Dunwoody Way, owned by developer and housebuilder Seddon Homes.
Post mortems concluded that both men died from crush injuries.
A third man, 45, who was also working on the crane, is recovering from serious injuries in Royal Stoke University Hospital.
In a statement on its website, Falcon said: “We consider health and safety to be of paramount importance and we have taken every step to ensure that all procedures are conducted as safely and efficiently as possible.
“We will leave no stone unturned to establish what went wrong and we will co-operate fully with all authorities to understand fully the causes of this tragic accident and to learn all safety lessons to be sure that safety remains of paramount importance.”
Cheshire Constabulary’s chief superintendent Matt Welsted said: “This is a multi-agency operation and we are working closely with the Health and Safety Executive, local authority, and building control to establish the full circumstances of the incident.”
He said the crane damaged a residential property when it collapsed but the occupants were not injured.
Bernard McAulay, national officer for construction at trade union Unite said: “Workers on construction sites throughout the UK will now have serious questions and concerns about the safety of similar cranes. It is imperative that the Health and Safety Executive investigates and as a matter of urgency provides advice and reassurance to ensure that similar accidents cannot and will not happen again.”
In March last year Falcon Crane Hire was fined £750,000 after a crane collapsed at a Barratt Homes development in Battersea, London, in 2006, killing two people.