Holbeche House Care Home, on Wolverhampton Road in Wall Heath, will shut within three months with bosses working to ensure a "smooth transition".
But GMB union chiefs have claimed some residents have been given 28 days' notice to leave and have branded the situation as "chaos" and "inhumane". Care home managers have refuted these claims, saying residents and staff have been given 90 days and they were doing all they can.
A protest was held on Tuesday at the care home, which offers residential, nursing and dementia care for up to 47 residents, by members of the GMB union.
Rachel Fagan, from the union, said: "This sudden closure has created chaos for residents and staff. Almost two dozen people – many who’ve lived in Holbeche for years – have been given just 28 days to move out. It's inhumane.
"Meanwhile workers are suddenly told they’re losing their jobs and being called into meetings, but not being told they can have their union rep present. The residents of Holbeche and those who care for them deserve better than this."
The historic Holbeche House site was once the hideout of Robert Catesby – and several members of the Gunpowder Plot were captured or killed there in 1605.
A spokesperson for the care home said: "Following a comprehensive review, we have made the difficult decision to close Holbeche House Care Home.
"The wellbeing of the people we support is always our absolute priority and this decision has not been taken lightly. Unfortunately, the home is no longer financially viable.
"Our senior management team is assisting the people we support, their families and colleagues during this process.
"We are working with the local authority close Holbeche House Care Home in 90 days. Our goal is to ensure a smooth transition for both residents and colleagues.”
Meanwhile a relative of a resident in the care home, who wishes to remain anonymous, said the situation has been "blown out of proportion" and the process was being handled efficiently and with care.
They said: "They had a meeting on Tuesday evening last week for family members and we were told that the home would be closing.
"They went into the background of why, obviously mainly the financial situation. I mean, it was all very good, there was lots of information, I think we were there for almost two hours really, talking over everything, and they told us that social services would be in touch to help us find somewhere else.
"Personally, it wasn't a surprise. I've never had an issue with the care there, the staff are lovely and my relative has always been very very well looked after. They quickly reassured us that it would be 90 days because they said they were aware of other homes that have closed very quickly, there was one that had closed giving residents only two weeks to find somewhere.
"So they were specifically quick to reassure us that it was 90 days. I don't know if that applies to staff as well. But for residential, it was definitely 90 days. My relative is quite excited to be moving somewhere new and closer to us, but one lady seemed visibly upset."
They added: "I can't speak for the staff, but the residents' family members, we've all been kept well-informed.
"And the company that's running the place at the moment has been very good and responsive. As a whole, things are being looked after.
"It's a sad situation and I'm sure the staff are very worried – and some residents are very worried because, you know, with dementia it can be very unsettling."
A spokesman from CQC said: "CQC inspectors are aware that the owners of Holbeche House in Dudley have taken the decision to close the service.
"We are monitoring the situation closely and liaising with Dudley Metropolitan Borough Council to ensure that people are moved safely to alternative services that are appropriate to their needs."
Councillor Patrick Harley, leader elect of Dudley Council, said: "The council was informed at the end of last month that the care home provider Holbeche House Ltd planned to close the home after they had been served notice by their landlord. We asked the care provider to fulfil their contractual notice period of 90 days, which they have agreed to.
"Our social workers and commissioning officers will support the residents, families and staff through this very challenging period. We will ensure all residents are safely transferred to alternate care which remains our top priority.
"At no point have we advised that any structural changes were needed at the property before new residents could be accommodated at the home. There were no formal restrictions on new placements and we had been working with the home to improve the quality of care for residents."