Barcelona plan to strengthen their defence this summer and Marquinhos is their primary target, with the Catalans willing to pay up to €50 million (£39m) to bring him to Camp Nou.
The intermediary Giuliano Bertolucci and Luan Aoas, Marquinhos' brother, met with Barca representatives and set out the 21-year-old's salary demands of €4.5m (£3.5m) a year plus a €3m (£2.3m) signing-on fee.
PSG do not want to sell Marquinhos and have twice refused to engage in talks with their European rivals, while the player and his girlfriend are happy and settled in Paris.
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There is a feeling at the club, however, that the former Roma starlet will consider a move if he continues to play second fiddle to Thiago Silva and David Luiz.
Barcelona would like to add a second defender in addition to Marquinhos, who is also wanted by Real Madrid and Chelsea.
Their shortlist includes Alessio Romagnoli of AC Milan, Valencia's Aymen Abdennour, Aymeric Laporte of Athletic Bilbao and Shakhtar Donetsk centre-back Yaroslav Rakitskiy.