At the age of 27, Asensio, who represents Spain on the international stage, will depart from Real Madrid as his contract comes to an end this month, becoming a free agent.
He won three Champions League titles in an eight-year spell in Madrid.
According to French media, Uruguay international Ugarte, 22, will cost PSG 60 million euros ($64 million). He has three years remaining on his contract with Sporting.
The two arrivals come days after Lionel Messi and Sergio departed the club and coach Christophe Galtier is also leaving.
PSG have been linked with moves for the French international trio of Bayern Munich defender Lucas Hernandez, Eintracht Frankfurt attacker Randal Kolo Muani and Borussia Moenchengladbach forward Marcus Thuram.
They are also in negotiations with Inter Milan’s Slovakia centre-back Milan Skriniar.