Skriniar joined Inter from Sampdoria in 2017 and made nearly 250 appearances in Serie A, the Coppa Italia and the Italian Super Cup during his time at the club. He also has 60 caps for Slovakia.
The defender was linked with a move to PSG in January but the two clubs did not work out a deal before the transfer window closed. Inter had reportedly rejected bids of more than 50 million euros or Skriniar last year.
The centre-back's final season in Italy was interrupted by injuries as he missed more than 20 games in all competitions, including the Champions League final when Inter lost to Manchester City.
"There is always pressure when you join a club like Paris St Germain. I feel it a little bit," Skriniar told PSG's website. "But I feel that I am ready, ready to be part of this very big club.
"I like pressure. I'm a defender so we know what it's like, you always have to be ready."
Skriniar is PSG's first signing of the transfer window, announced a day after the club appointed Luis Enrique as manager to replace Christophe Galtier.
Former Real Madrid attacker Asensio was signed on a three-year deal until 2026, PSG said.
The Spaniard, who broke out under Zinedine Zidane in the 2016-17 season and scored in Real's 4-1 win against Juventus in that season's Champions League final, had gradually fallen out of favour at the LaLiga club over the last few years.
The forward made 51 appearances for Real across all competitions last season, scoring 12 goals, but was used mainly as a substitute.
Skriniar and Asensio are PSG's first signings of the transfer window, announced a day after the club appointed Luis Enrique as manager to replace Christophe Galtier.
