Kurzawa moved to the Ligue 1 leaders from Monaco in the summer of 2015.
The 25-year-old has since made 70 appearances for the Parisians, but Sport say that he is now ready to leave the Parc des Princes having grown frustrated with his lack of regular playing time under Unai Emery.
Kurzawa has started only 12 Ligue 1 games so far this season after the Spanish boss bolstered his left-back options in the summer with the signing of compatriot Yuri Berchiche from Real Sociedad.
Sport add that Kurzawa has recently parted ways with his agent and hired replacement Moussa Sissoko with the clear mission of getting him a new club ahead of the summer transfer window.
The Spanish publication says that Barcelona could consider the proposal as Ernesto Valverde is looking for a new left-back ahead of next season to replace Lucas Digne.
Furthermore, Sport point out that luring Kurzawa away from PSG would only cost around €25m (£22.1m, $30.7m) and understand that it could be a reasonable fee for the La Liga giants given the current inflated market.
Barcelona only signed Digne from PSG in the summer of 2016. The Frenchman had enjoyed a positive loan spell at AS Roma during the previous season and was expected to provide competition for Jordi Alba at left-back following the departure of Adriano to Besiktas. However, the 24-year-old has since failed to make the expected impact at the Nou Camp.
Indeed there were already suggestions last summer that Barcelona were considering partying ways with Digne when the Catalans battled with Real Madrid for the signing of Theo Hernandez from Atletico Madrid.
Theo finally moved to the Santiago Bernabeu and Digne stayed at Barcelona to serve as back-up to Alba.
However, he has since made only 15 appearances under Valverde and Sport say that the Catalans could show him the exit door in the summer to make space for a new left-back.