The city council has proposed cuts in seven areas of community services, including withdrawing some funding to outside care organisations.
It means about a third of its planned £63m savings over three years would be in Health and Community Services.
The council said it could still help the city's "most vulnerable people".
The Unite union said industrial action would be considered if its members considered the changes "a step too far".
Ann Lucas, cabinet member for Health and Community Services, said the savings resulted from "huge cuts" in government funding to the council.
The Labour-run council's biggest directorate covers adult social care, public safety, housing, libraries and adult education.
Ms Lucas said although the directorate was looking to save £15m over three years, increasing demands on care made the figure closer to £22m.
"The seven areas we are looking at means we could make the savings we need over the next two years while continuing to protect and help the most vulnerable people in the city when they need us," she said.
"But it is important to stress that no decisions have been made and that after consultation, we may not press ahead with the proposals."
However, she said if the plans did not go ahead, savings would be sought in other areas.
The council starts formal consultations with its partners over the changes in June.
The plans include:
Alan Lewis, from Unite, criticised the proposed cuts to jobs and services, adding "I dread to think of the impact it will have on vulnerable people".
"I can see from the proposals straight away it will lead to bed blocking in hospitals because there will be no support for people to get them back and into their homes," he said.
About 200 carers of elderly residents and people with learning disabilities in Coventry could lose their jobs under council proposals.