James, who last week signed a six-year contract extension tying him to his boyhood club until 2032, picked up the issue during Chelsea's 1-0 Premier League home defeat by Newcastle United.
The 26-year-old England international did not take part in training ahead of the Stamford Bridge game, with Chelsea needing to overturn a 5-2 first-leg deficit against the reigning European champions.
"Reece felt something in his hamstring towards the end of the Newcastle game," Rosenior told reporters. "We don't know the full extent of the injury yet, but he will be out for tomorrow's game.
"It could be serious - with a hamstring injury, it's never ideal. Hopefully, we can get him scanned and find out more."
Rosenior also provided updates on other fitness concerns, with right back Malo Gusto and goalkeeper Filip Jorgensen both doubts.
"Malo is ill today, so we'll look at that tomorrow. With Filip, he's felt something in his groin and we will get him scanned," he said.
Chelsea received a boost on the disciplinary front, with winger Pedro Neto cleared to play after UEFA handed him a warning for pushing a ball boy during the first-leg defeat at the Parc des Princes.
"He's a fantastic player, he has apologised for the incident and I think UEFA have made the right decision," Rosenior said.
Earlier on Monday, Chelsea were fined a record total of 10.75 million pounds for historical breaches of Premier League regulations. The club were also handed a one-year first-team transfer ban, suspended for two years, along with an immediate nine-month academy transfer ban.
Rosenior played down the impact of the sanctions on preparations for Tuesday's match.
"It's not a negative distraction. It's a line drawn through that issue. (Now) we can plan to make this club as strong as possible," the 41-year-old Englishman said.
Despite the injuries, off-field issues and a daunting scoreline, Rosenior insisted his side still believe they can trouble PSG.
"I think this week was difficult for us. We know our strengths, we still believe because we showed in the Club World Cup when we won 3-0 (against PSG in the final). The group believe, the manager believes, the fans should believe as well," he said.