Mrs May's predecessor, former prime minister David Cameron had promised to bring in an upper limit of about £75,000 on the amount people must pay towards their own care.
But this met with delays and there are now rumours the cap will not be introduced until well into the next decade.
Peter Wright, director at IFA Plan Money, said Aegon's findings were similar to preferences expressed by clients.
He said: "Broadly speaking, our client’s views reflect the findings of Aegon’s report. Most clients and their families accept that they should fund the majority of care costs but believe there should be state support too.
"The family home is the most contentious issue, particularly when this is the only real asset owned. For many people the family home is sacrosanct and should be viewed separately to other assets which have been acquired as investments."