Hull City Council said it needed to make some of its services "financially sustainable" following cuts to its central government grant.
The authority said it would "remodel" learning disability services, older people's care homes, day care and support for people leaving hospital.
Sarah Littlewood, of the union Unison, described the cuts as "devastating".
"The need is not reducing," she said.
"The need, in fact, is increasing. So, for the people who receive those services and for the people who deliver that those services, it is absolutely devastating that more cuts are on the agenda."
The authority said the £15.3m cuts to its £95 million adult care budget was in addition to the £10m it had removed since 2010.
Councillor Stephen Brady, Labour leader of Hull City Council, said: "The government has imposed deep funding cuts on adult social care, despite the fact that we have an ageing population, creating an even greater need for care.
"We now have no option but to make further cuts but we are determined to do this in a way that will ensure we are able to continue to offer a high standard of care to some of the most vulnerable people in our community."
The plans will be put out to public consultation over the summer.
Adult social services in Hull will face cuts of more than £15m over the next five years.