Casado saw his playing time reduced last season compared to his debut campaign, and the picture looks set to worsen with Gavi and Fermin Bernal both due back at full fitness under Hansi Flick.
According to Marca, the club and player agreed several weeks ago that a departure was the best route forward, with the message passed on to his agent Jorge Mendes to find a new club.
Several proposals are said to be on the table, but Al Hilal are currently applying the most pressure and have tabled the best offer.
Casado is understood to be uncertain about a move to Saudi Arabia, aware of the financial rewards on offer but less convinced it represents the right sporting decision, though he has reportedly grown more receptive to the idea in recent days.
Al Hilal are not the only side in the running, with AC Milan among the European clubs also weighing up a move for the midfielder.
Atletico Madrid were initially interested but are understood to no longer be as convinced over signing him.
Barcelona had hoped to complete the sale before 30 June in order to include it in the previous season's accounts, but have opted to give themselves more time in the hope offers improve.
The club's sporting management believe they could recoup around €30million in fixed and variable fees, particularly if the Saudi Arabian option materialises.
Barcelona will look to close the deal this week, with the squad due back for pre-season training on Monday 13 July, though resolving it before then is not considered essential.
Away from transfer business, Barcelona are also restructuring their fitness department, creating a new Performance and Optimisation Area to unify physical preparation methodology across every section of the club, not just the first team.
Flick had been pushing for change in this area after a difficult season on the fitness and recovery side, most notably with Raphinha, who suffered a relapse of injury and was sidelined for a month longer than expected.
That push has now taken shape: Julio Tous, previously the club's head of first-team fitness, moves up to lead the new club-wide department, overseeing methodology across all of Barcelona's sections.
The specific first-team fitness role he leaves behind will instead go to Yann-Benjamin Kugel, brought in personally by Flick as his own trusted appointment to fix the exact issues that prompted the shake-up.
Flick and Kugel worked together for several years at the German Football Federation, and it is that relationship of trust that led the coach to push for his arrival.
Kugel spent a decade at the German federation, working as part of Joachim Low's coaching staff during the 2014 World Cup win, and has also held fitness roles at Cologne, Werder Bremen, PSV Eindhoven and Benfica, as well as with the Israel national team.
He arrives at Barcelona with the specific first-team brief of reducing injuries, one of the club's biggest issues last season, and preparing players to sustain the high-intensity pressing that defines Flick's teams for the full 90 minutes.