The deal will see a building once dubbed the ugliest in the city by the local paper torn down and turned into a private apartments and a four star hotel complex for client Rex Developments.
Carillion has been head to head with Norwest Holst for the deal over the past few weeks but has emerged as the front runner. Laing O'Rourke and Willmott Dixon also looked at the job but decided to not tender.
One insider said: 'We have heard that Carillion, who originally went for the job as Mowlem, has been successful and it looks like the contract may be officially signed in the next few weeks.'
The pair will use the time to determine a final cost for the job.
Located on Broad Quay in the city centre, the 18-storey block was formerly home to the Bristol & West Building Society. The new development will involve rebuilding the tower using the shell as a frame.
Local firm Wring Demolition has been on site for the past few months stripping the building and carrying out enabling works.
A source close to Wring said: 'We are set to be completed on site at the end of June and we know the client is hoping to start construction work pretty much straight away. We understand they are close to official contract award.'
The hotel will have 178 bedrooms while the 168 residential apartments will be accompanied by shops, cafes and restaurants.
The plans will retain Seven Broad Quay, a Grade II listed building with historic walls on Marsh Street. Other buildings surrounding the tower will be demolished and rebuilt. Construction work will start in July.
Scheduled for completion in 2007, the project has been drawn up by Bristol architect Atkins Walters Webster.