Paul Anthony Clough was hit by a car as he was crossing Wood Street, near the junction with Monument Road, shortly after 10pm on Tuesday, April 20.
The 46-year-old was rushed to hospital with head and leg injuries and remains in critical condition, police said.
Officers say they are struggling to trace Mr Clough's family and have urged anyone with information to get in touch.
Sgt Mark Crozier, from the Serious Collision Investigation Team, said: "Mr Clough remains in a critical condition in hospital so it's important that we find his family and update them on his condition.
"I would urge the family of Mr Clough to get in touch or anyone who has any information on who his next of kin may be."
Anyone with information has been asked to get in touch via Live Chat on the West Midlands Police website or by calling 101, quoting log number 4688 of April 20.
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