Bearing in mind Florida's vast elderly population, Mr Perry said he advocated reform of Social Security, while adding that it should be safeguarded for existing beneficiaries.
But Mr Romney pointed to Mr Perry's past criticisms of Social Security.
"There's a Rick Perry out there that's saying that... the federal government shouldn't be in the pension business, that it's unconstitutional and to be returned to the states," said Mr Romney. "So you better find that Rick Perry and get him to stop saying that."
Mr Perry said his stance on the issue had shifted "not one inch, sir", then hit back saying that Mr Romney had changed his position on healthcare between editions of a book he had published.
In one edition, Mr Perry said, Mr Romney had advocated expanding the healthcare programme he signed in Massachusetts to the rest of the country.
"Then in your paperback you took that line out, so speaking of not getting it straight in your book, sir," said Mr Perry.
"I think Americans just don't know sometimes which Mitt Romney they're dealing with," Mr Perry added.