The 26-year-old right-back picked up the problem towards the end of Saturday's 1-0 Premier League defeat to Newcastle United, with the club now awaiting scan results to determine the full extent of the setback.
"Reece felt something in his hamstring at the end of the game against Newcastle," Rosenior said. "It's really frustrating and a disappointing one for us. We don't know the full extent but it rules him out for tomorrow."
When asked whether the injury could keep James out for a number of weeks, Rosenior admitted that was a possibility.
"With a hamstring injury, it is never great and for Reece we can hopefully get him scanned, find out the full extent and then we will know more because we know how important he is and what a leader he is in the group," said the Blues boss.
The injury is another blow for the England international, who has struggled with hamstring problems in recent years. It is his 10th hamstring injury since December 2020, although he had largely stayed fit this season after careful management by the club.
James had recently spoken about finally enjoying a consistent run of games.
"I had to come through a lot, two or three years of struggles and setbacks," he said last week after signing a new six-year contract extension that will keep him at the club until 2032. "I'm in a really good place at the moment. My body is feeling good, I'm playing on a regular basis, two or three games a week."
His latest setback could also affect his chances of being involved in England's upcoming internationals later this month, with Thomas Tuchel set to announce his squad on Friday.
Chelsea are also dealing with several other fitness concerns. Goalkeeper Filip Jorgensen is undergoing a scan after feeling a groin problem, while right-back Malo Gusto missed training due to illness.
"Malo is ill today so we'll have a look at him for tomorrow," Rosenior said. "Filip felt something in his groin after the PSG game and again we're getting him scanned to find out the full extent of that injury."
There was more positive news elsewhere in the squad, with Estevao Willian returning to training after a hamstring injury and pushing for a return.
Meanwhile, winger Pedro Neto remains available after receiving only a warning from UEFA following an incident involving a ball boy during Chelsea's recent Champions League clash with Paris St-Germain.
"There was no real bad intention, it was just a will to get on with the game so I think UEFA made the right decision," Rosenior added.
Chelsea have been handed the largest fine in Premier League history and a suspended transfer ban after admitting to historical financial rule breaches involving more than 47 million pounds in undisclosed payments during the Roman Abramovich era.
Chelsea's pursuit of a place in next season's Champions League has entered a pivotal phase, and now head coach Liam Rosenior has found himself under increased pressure.
Eddie Howe praised Newcastle's "defensive mentality" after his side secured a hard-fought 1-0 victory over Chelsea at Stamford Bridge on Saturday.
Chelsea head coach Liam Rosenior has criticised referee Paul Tierney after a bizarre pre-match moment saw the official standing in the middle of the Blues' team huddle before their clash with Newcastle United.
