The 28-year-old has suffered a "strained interior medial ligament" in his right knee, PSG said in a statement.
Italy kick off the delayed European Championship on June 11 against Turkey at the Stadio Olimpico in Rome.
Verratti will miss the rest of the domestic season in France, where PSG trail leaders Lille by three points with two Ligue 1 matches remaining and face Montpellier in the French Cup semi-finals on Wednesday.
Verratti's last appearance at a major tournament was in the 2014 World Cup, as Italy failed to qualify for the event four years later.
He also missed Euro 2016 through injury.
He has spent two spells on the sidelines this season, once with a thigh injury and another after contracting Covid-19.
"He is very disappointed, we all are," said PSG coach Mauricio Pochettino. "We are losing an important player for the rest of the season and the crucial matches.
"He is already thinking about recovering as quickly as possible. Let's hope that he is back in time to participate in the Euro with his national team."
At Euro 2020, Roberto Mancini's Italy also face Switzerland on June 16 and Wales on June 20 in Rome in Group A.
Verratti has scored three goals in 40 international appearances since making his debut in 2012.