Saliba was substituted during France's semi-final defeat to Spain and could require surgery on a fracture in his back, having reportedly sustained the injury during a Premier League match roughly two months earlier and continued to play through it.
Arsenal, who are defending Premier League champions, are yet to confirm a timescale for his return, though French media reports suggest a lay-off of four to five months is likely.
Deschamps, speaking ahead of France's third-place play-off with England, confirmed the extent of the problem.
"William Saliba got injured and it's problematic," he said. "He went through some examinations. The injury did not worsen.
"He faced that injury since March. He had some pain and unfortunately the pain was not bearable."
Saliba had continued to play through the discomfort for both club and country, managing the issue with painkillers and a modified training programme overseen by Arsenal and France's medical staff. He started France's opening two group games before being rested for their final group fixture against Norway, then returned for the knockout rounds before going down without contact early in the semi-final and being replaced by Maxence Lacroix.
The centre-back's injury is a significant blow to Arsenal as they prepare to defend their Premier League title, with the 25-year-old having played 31 of their 37 league games last season. Arsenal won only 50 per cent of matches in his absence last season, underlining his importance to Arteta's defence. Mikel Arteta's side, who also reached last season's Champions League final, will now assess options to cover his absence, with Cristhian Mosquera in line for an extended run in the side.
Saliba is understood to have told team-mates his back "was gone" as he was helped from the field in Dallas. Further scans in the coming days are expected to determine whether surgery goes ahead, with Arsenal preparing for the possibility of starting the new Premier League season without one of their key defenders.