Barca tried to bring the 27-year-old back to the club over the summer transfer window, although were ultimately unable to agree a deal with the Ligue 1 champions.
Injuries to Lionel Messi and Ousmane Dembele opened the door to 16-year-old Fati to get a chance in the first team - an opportunity he's grabbed with both hands.
In a hugely impressive start to life at Camp Nou, Fati has already scored twice in the first three La Liga appearances of his career but Rivaldo has urged supporters not to expect too much, too soon.
He told Betfair: "As I've said before, the start of the season is a great time to try out new players. The lack of decisive matches allows some error margin for the youngsters, and [Ernesto] Valverde is doing it very well in Barcelona. He has already been rewarded with Ansu Fati showing huge talent on his first appearances.
"The 16-year-old is already showing he has the personality to play big matches and made his Champions League debut this week in the 0-0 draw with Dortmund - a match in which Barcelona didn't play well.
"If Barcelona had been playing well perhaps the youngster would have shone even more, but that wasn't the case this time.
"I understand fans' enthusiasm about the young Ansu Fati but it is far too early to start comparing him to Lionel Messi and wondering if he may be his future replacement.
"No one doubts Fati's big talent, but it's premature to put so much responsibility on his shoulders when he is still a kid. Let's give him time and space to play his normal football, and the future will tell if he is about to become the 'new Messi' or not.
"There are other rumours mentioning that Fati's breakthrough will lead Barcelona to give up on signing Neymar. Honestly, I think it would depend on teams' results until the next transfer window.
"If Barcelona manage to play well, with Fati being an important part of the success, they have no reasons to sign a new player. But if things are not going so well it can be different, and in that case Neymar could become a target again."
Fati and Barca were held to a 0-0 draw by Borussia Dortmund in their Champions League group-stage opener and Fati will be hoping for more minutes when the Catalans face Granada in La Liga on Saturday.