A number of companies are understood to have expressed an interest in buying the 56-year-old contractor, which has remained in the Sims family for three generations.
It is understood that Carillion has emerged as favourite to acquire the company. Try Construction, which made enquiries about acquiring the company in the immediate aftermath of its collapse, has pulled out.
Southwark council is understood to have taken over Sims' contract to renovate the Canada Estate in Southwark.
The fate of Sims' other uncompleted contracts is still undecided at the time of going to press.
The company has officially served notice on its 85 employees. Sources close to the company said it was unlikely that any of the workers would be re-employed by a new owner.
An industry source said: "The company is still a good buy for any firm that wants to expand its client base and profile in the South-east, but that won't necessarily involve anyone presently at the company. Until now, they had an excellent reputation."