The midfielder, whose current deal ends next year, was not offered a new contract with the Magpies this summer after talks broke down in May.
Barton has indicated that he will see out the remainder of the contract before leaving on free transfer, but manager Pardew has said that he could have a change of heart.
“Joey sometimes has an opinion that can change,” he told BBC Radio Newcastle. “I think the most important thing for Joey is that he can see the squad’s got real quality in it.
“A few times last year he came to see me frustrated we weren’t getting the results in terms of the performances he was putting in, and I had to agree with some of that.
“I spoke to Joey over the summer and I think me and him have got a good relationship. If I can keep him where he was last year for another year, at the end of the season who’s to say we can’t agree a contract?
“Joey needs to see that success on the pitch and if we give that to him, I think he could stay.”
The 28-year-old moved to St James’ Park from Manchester City in 2007 in a £5.8m deal.