The review led to fears that the project, which would also create facilities for Imperial College and a building for the National Heart and Lung Institute, would be scaled down.
As part of the review the two trusts running the scheme, the Royal Brompton and Harefield and the St Mary's, were asked to submit a report putting the case for the scheme to go ahead. A source close to the hospital said this report was due to be handed to the government on Tuesday.
The source said that the review team, which includes representatives from the Treasury, the Department of Health and the National Audit Office, could reach a decision next week.
Sources close to the Treasury and the client expressed optimism that the scheme would receive the green light.
One said: "We are expecting an announcement next week. The feeling is more positive than negative right now."
Another source close to the Treasury said tighter parameters had been set. He said: "There are more closely defined boundaries for the scheme.
There is a feeling it got a bit out of control, which may or may not be true
Source close to the project
The source said: "For a myriad reasons it is a complex scheme. There is a feeling that it got a little bit out of control, which may or may not be true. But perception is everything in politics."
The inquiry is also looking at a plan that the two trusts put forward to lease a nearby office building to which staff could be relocated during construction.
One source claimed that the trusts might be permitted to use two of the floors of the building, which is part of the Paddington Basin development.
The review was set in motion after contractors raised concerns late last year about the capacity of the market to take on such a large scheme.