Sage is under pressure after a difficult start to his first season at Selhurst Park, with six defeats in nine Premier League matches leaving Palace in the relegation zone.
The Eagles have two important fixtures before the international break, hosting Hoffenheim in the Europa League on Thursday before welcoming Liverpool to south London on Sunday.
Those results could prove significant for Sage's position, with Palace already considering possible alternatives should they decide to make a change.
Barry has emerged as a frontrunner and has reportedly been sounded out by club officials about the possibility of taking charge, reports the Mirror. The 40-year-old has never managed a Premier League side, meaning the role would represent his first opportunity as a top-flight head coach.
He has, however, built a strong reputation through his work at club and international level. Barry previously worked at Chelsea and Bayern Munich, as well as alongside Roberto Martínez with Belgium and Portugal.
His relationship with Thomas Tuchel has been particularly influential in his career. Barry worked under the German at Chelsea before following him to Bayern, and was subsequently appointed to Tuchel's England coaching staff.
The pair have since extended their agreements with the national team until the 2028 European Championship.
Barry's coaching experience has attracted interest from clubs looking for a new manager, while the opportunity to take charge of a Premier League side could be an appealing next step.
For Crystal Palace, the immediate priority remains deciding whether Sage can turn around the team's results. The Frenchman was appointed in the summer after leaving Lens, but his first months in England have proved difficult.
The club's position could become clearer after the matches against Hoffenheim and Liverpool, with the latter providing a significant test before the international break.
Barry is understood to be among the options being considered, rather than having been appointed or formally offered the job. Any move would also require a decision on his future with England, where he remains part of Tuchel's coaching team.