The Dutch starlet attempted to continue after picking up the injury in the seventh minute of the 2-0 win at the County Ground, but Ramires eventually replaced him halfway through the first 45 minutes.
The report claims he could miss the entire campaign depending on the severity of the ligament problem.
An £8m summer signing from Vitesse Arnhem, van Ginkel had been earmarked for an important first-team role at Stamford Bridge this season.
But Chelsea assistant first-team coach Steve Holland admitted after the match that the initial signs are not positive.
"It's too early to say, but it doesn't look good for Marco," said Holland. "We'll have to wait for the results of the scan, but we have at least two casualties from this evening."
The second injury was suffered by Ramires - van Ginkel's replacement - who was forced off at half-time and is now a doubt to face Tottenham at White Hart Lane this weekend.
The Daily Mail reports that van Ginkel has returned to the Chelsea training ground in Cobham for treatment, where he'll also have a scan later today.
Jose Mourinho's side progressed to the fourth round of the Capital One Cup with a 2-0 win over the Robins - the goals scored by Fernando Torres and Ramires.