The Netherlands international was substituted after just 10 minutes on Tuesday evening after a challenge by Swindon midfielder Alex Pritchard. The 20-year-old initially played on following the tackle but hobbled off minutes later.
Van Ginkel, who joined the club from Vitesse Arnhem this summer for 8 million pounds, was making just his second start for the club in the Capital One Cup fixture.
After the game, Chelsea assistant first-team coach Steve Holland said of the injury: "We've got an injury to Marco, which, it's a little bit too early to be certain, but it doesn't look good.
"He'll have a scan on Wednesday on an injury to his knee. It's too early to say (if it's a bad one) but let's hope that's not the case. It didn't look good as he came off so early in the game. We'll have to see the results of the scan, we'll know more on Wednesday."
Chelsea also suffered an injury blow to Brazilian midfielder Ramires. His injury is considered to be less serious than Van Ginkel's, though he is expected to miss the weekend fixture with Tottenham.