Kier was unveiled as preferred bidder on the design and build job last September.
But contractor and client are understood to have been unable to agree a final price opening the way for rival bids.
One source close to the project said: “It seems Kier wanted more to build the tower so the door was suddenly opened to others.
“We had a look at the job but the word is that Ardmore has now picked it up.
“Quite a few big names are getting cold feet around town on bids they have been involved in so the way is clear for some other names to start picking up prestigious jobs like this one.”
Work on the contract – known as M103 -will involve construction of 224 luxury apartments arranged in four blocks, anchored by a 24-storey tower.
The Greenwich Peninsula development is a major regeneration scheme which will include over 160 acres of residential, business and leisure space with nearly two miles of river frontage.
In total more than 60 new development plots will be created by Knight Dragon by 2039 and the scheme will deliver over 10,000 new homes for up to 23,000 residents.
Wates and GB Building have already secured major building projects at the Greenwich Peninsula site.
But Willmott Dixon walked away from two tower blocks on the project 12 months ago after failing to agree terms with Knight Dragon.