Vieira has resigned as the manager of Major League Soccer side New York City.
The 41-year-old is set to take over at Nice, replacing Lucien Favre who has become the new Borussia Dortmund boss.
Vieira was contacted over the Arsenal job last month but discussions did not go past an early stage.
Unai Emery was instead given the responsibility to replace Arsene Wenger at the Emirates.
Parlour believes it is important for Vieira to succeed at Nice and is backing him to return to England if he does.
“Patrick, my old team-mate, is going to Nice, which is a good little move for him,” Parlour said on talkSPORT.
“It’s lovely down there, beautiful, south of France.
“I think this is a stepping stone. He’s got to do well here though, that’s the most important thing for Patrick.
“He’s learnt from New York, he’s moving on.
“It’s a little bit more of a test for him I should imagine in Nice.
“Let’s see how he does because I’m sure one day he would love to be back in the Premier League and I’m sure he will be one day.”