The former Arsenal and Juventus goalkeeper came out of retirement to sign a one-year contract with Hansi Flick's side in October after first-choice Marc-Andre ter Stegen suffered a season-ending knee injury.
However, Inaki Pena has kept Szczesny out of the starting lineup for the time being, and the 34-year-old has only been an unused substitute since making the move to Catalonia.
As things stand, Szczesny will become a free agent again at the end of the campaign, and he will be free to negotiate a pre-contract agreement with foreign clubs from January onwards.
Alternatively, the ex-Poland international could decide to hang up his gloves again when the campaign comes to a close, but according to Marca, the club want to reward his work ethic with a new deal.
Szczesny to earn new deal thanks to 'exemplary' attitude?
When he arrived at Camp Nou, Szczesny apparently accepted that he would be playing backup to Pena, and he would therefore be demoted to third choice if Ter Stegen returned before the end of the season.
Nevertheless, the Barcelona powers-that-be are said to have been left thoroughly impressed by Szczesny's 'exemplary' attitude on the training pitches, as he is working at full throttle in case he is ever called upon.
Both head coach Flick and sporting director Deco are therefore keen to prolong Szczesny's contract until the summer of 2026, assuming that the goalkeeper continues to train in the same vein for the remainder of the season.
Furthermore, the duo are making contingency plans in case Pena decides to move on next season, as if Ter Stegen immediately becomes first-choice again after his recovery, Pena may look to join a club where he is the undisputed number one.
Deco and Flick are supposedly keen to have three high-quality goalkeepers in the ranks, though, and they will therefore sit down with Szczesny to discuss a short-term renewal in the near future.
Deco: 'The Barcelona dressing room is fond of Szczesny'
The Barcelona sporting director recently waxed lyrical about Szczesny to Mundo Deportivo, revealing that the Camp Nou dressing room are 'fond' of the former Gunners' shot-stopper.
"Szczesny hasn't played yet. The first thing he came to do was to bring balance and competitiveness. He's a person who the dressing room is quite fond of. He's come with a positive spirit and that's important for the dynamics of the dressing room," Deco said.
"We're happy with what he's doing. The decision to play will always be up to the coach, but I think competitiveness in the squad is important. What we want with Szczesny is for him and Inaki to be fighting to play and for both of them to be working and respecting each other."
Szczesny won two FA Cups and one Community Shield with Arsenal before achieving the zenith of his career in Italy, where he became a three-time Serie A champion, three-time Coppa Italia winner and two-time Supercoppa Italiana champion.
Barcelona hold a six-point lead over Real Madrid at the top of the La Liga table and return to action away to Celta Vigo on November 23 after the international break.