Pat Garbutt said that the local authority had had to intervene after four privately-run care homes in the district recently closed.
Councils have a legal responsibility to find residents new homes if the venue they have been living in is shut down.
The Care Quality Commission (CQC) has placed several homes in the district in special measures this year.
In that scenario, the home’s management is told to address issues raised imminently or face losing its licence to run.
Speaking at a health scrutiny committee meeting on Thursday, Coun Garbutt said: “We have had four closures of care homes in the last four or five months. It’s an area we are concerned about.
“Sometimes very short notice is given to us – we could be given a week – to find new places for the residents.
“We’ve got to deal with it sensitively because the people in the homes want to be near their families. That’s been one of the most difficult positions we’ve been in.”
Scrutiny chair Betty Rhodes described the picture of care homes in the district as “disappointing”.
She said the committee would write to Coun Garbutt and the council’s health department to outline its concerns.
Coun Rhodes said: “I understand that for care homes costs have been rising.
“It’s difficult enough to deliver the care that’s needed for individuals in the home.
“But it must be very difficult for the relatives of people in a home that needs to be closed down.
“It’s a traumatic situation for everybody concerned.
“This authority has a responsibility (to residents in care homes) and we will make sure that that responsibility is taken seriously.”