PSG signed Neymar for a world-record €222 million (£200m/$262m) in the summer of 2017, and he has won two Ligue 1 titles as well as the Coup de France in his two seasons in the capital.
The Champions League has eluded him, and Thomas Tuchel’s side were knocked out by Manchester United in the round of 16 last season.
The former Santos forward has been linked with a move away from PSG this summer, with Real Madrid a rumoured destination.
Barcelona last won the Champions League in 2015, when Neymar scored with virtually the last kick of the game to give the Catalans a fourth title in 10 years and fifth in their history with a 3-1 victory over Juventus.
The Spanish giants suffered a humiliating exit from the Champions League in the semi-final this season, blowing a 3-0 first-leg lead as Liverpool stunned them with a 4-0 win at Anfield to book a place against Tottenham in the final.
On his Instagram acccount, Rivaldo said signing Neymar “would be a great recruitment” for Barca “for everything that happened in this season’s finale”.
He added: "He is, in my humble opinion, the player who Barcelona lack to be champion again in the [Champions League]. He has personality and a lot of quality to win back all the fans of Barcelona and also to be the best in the world.”
Neymar will put thoughts about his club future on hold as he aims to help Brazil win a first Copa America crown since 2007.
They didn’t get out of their group in 2016, but Rivaldo is confident about his country’s chances ahead of the kick-off against Bolivia in Sao Paolo on June 14.
He said: “I think Brazil will win the Copa America with a certain ease. I do not see any selection that can beat Brazil mainly with the return of Neymar who, for me, will be the best player in the competition.”