Ruben Amorim's struggling United travel to face the French side on Thursday in the first leg of their Europa League quarter-final.
United are 13th in the Premier League after a miserable season but remain unbeaten in Europe.
Onana said after the Manchester derby on Sunday, which finished 0-0, that overcoming Lyon, fifth in Ligue 1, was "not going to be easy, but I think we are way better than them."
That bullish remark was put to the normally understated Matic on the eve of the first leg in France.
"You know, I respect everyone, but to say that, then you need to give the answer," said the Serbia international, who also had a spell with Chelsea in the Premier League.
"If you are one of the worst goalkeepers in Man United's history, you need to take care of what you are talking (about).
"If David de Gea or Peter Schmeichel or (Edwin) van der Sar said that, then I would question myself.
"But if you're one of the statistically worst goalkeepers in Man United's modern history, then he needs to show that before he says (anything). We will see."
Onana then posted a response on X, having a dig at Matic, who did not win a trophy during his time at United, though he collected silverware while he was at Chelsea.
"I would never be disrespectful to another club. We know that tomorrow will be a difficult game against a strong opponent," said the Cameroon international.
"We focus on preparing a performance to make our fans proud.
"At least I've lifted trophies with the greatest club in the world. Some can't say the same."
United paid around £47 million to sign Onana from Inter Milan in 2023 as a replacement for De Gea, and he won the FA Cup in his first season.
Only two goalkeepers have kept more clean sheets than his total of nine this season in the Premier League, and he is capable of spectacular saves.
But the 29-year-old has made several costly errors during his United career, and his reliability continues to be questioned.
Despite his sharp words about Onana, Matic said he "will always keep United in my heart" and hopes they eventually turn things around under former Benfica teammate Amorim.
The midfielder is grateful to the United manager for helping him settle in Portugal and believes he can succeed at Old Trafford with the right support.
"In football today, people don't have time, but I'm sure he's the right profile," he said. "He's a young coach with lots of energy. He needs support.
"It's hard in the Premier League if things are not right. It's so difficult to turn things (around) in a short period, but step by step, I believe that he can do very good. But I hope that he will not start tomorrow."
The dour 0-0 draw with City at the weekend left United languishing in 13th place in the Premier League - on track for their worst league finish for at least 35 years.
The Europa League offers them their only route back into Europe next season. Failure to win the trophy in Bilbao next month would have severe repercussions for a club already struggling financially.
