Work on the scheme has stalled following the collapse of industry giant Carillion in January.
With the project delayed until at least 2022, it is expected that the hospital will cost at least a further £150 million to complete.
The Trust initially stated that it would seek to complete the project under a Private Finance Initiative 2 scheme. However, it has been unable to find an interested contractor, while the original financial backers have abandoned the development.
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As a result, the Trust has now called upon the Government to directly fund the completion of the hospital.
Unite's acting West Midlands regional secretary, Howard Beckett, said the Government must "stop dithering and cough up the money to finish the project".
"Any alternative scenario is purely about dogmatic ideology rather than doing what is best for patients and workers in the West Midlands," he said.
"The West Midlands economy needs the project to be restarted as soon as humanely possible to begin rebuilding the damage caused by Carillion's demise.
"It also must never be forgotten that patients and local residents desperately need a new hospital to ensure they receive the healthcare they deserve, these ongoing delays are denying them that basic right."
(LM)
