A bitter row has broken out between the two after Multiplex last week gave Cleveland Bridge 28 days to leave the site. The Dutch firm Hollandia is to be parachuted in to replace it.
Cleveland Bridge has warned that the move may lead to delays in the £757m project.
A statement issued by the Darlington-based firm said: "The uncertainty surrounding staff and labour may introduce delays to the overall project." However, it added that "Cleveland Bridge remains committed to executing the remaining parts of its work on time and delivering this extraordinary project for Britain".
Multiplex rejected any suggestions that the project faced delays.
A source at Cleveland Bridge said the firm was unhappy with the way that Multiplex had treated it, and was disappointed to be removed from the contract.
The uncertainty surrounding staff and labour may introduce delays to the overall project
Cleveland Bridge statement
He claimed that Cleveland Bridge had been given the steelwork package with a deadline of August 2005. He said Multiplex had recently demanded that this be moved forward to June 2005.
He added that senior managers at Cleveland Bridge were considering legal action against Multiplex.
A spokesperson for Multiplex said that any suggestion that deadlines had been changed was "absolutely untrue". He said: "Deadlines have always remained the same."
The fate of Cleveland Bridge's 200 employees on the site was unclear as Building went to press on Wednesday.
Cleveland Bridge had been in talks with trades unions Amicus and the GMB with a view to transferring its staff to Hollandia. However, these efforts appear to have failed.
The firm's statement said: "Cleveland Bridge regrets to announce that it is unable to reach agreement with Multiplex on the transfer of Wembley site staff and workforce to Hollandia, the Dutch company chosen by Multiplex to complete the steelwork."
The statement went on: "Cleveland Bridge therefore has no alternative but to give notice of redundancy to its entire Wembley staff."
Multiplex's spokesperson said talks about the fate of Cleveland Bridge's staff were continuing.