Odobert came off in the 35th minute at the Tottenham Hotspur Stadium after landing awkwardly following an attempted tackle on Harvey Barnes.
The French winger received treatment on the pitch before walking straight down the tunnel when replaced by Mathys Tel.
Frank was unable to offer a clear update on Odobert’s injury, but hoped it was not serious with Spurs already short of options on the wing.
“No \[update\], not yet,” Frank told reporters at full-time.
“I hope it's nothing too serious, but I don't know what it can be. Of course, I hope it's not a serious injury to a 20-year-old young player.”
Spurs slipped to within five points of the relegation zone as their winless run in the league extended to eight games.
Frank, who met with the Spurs hierarchy on Monday, insisted he was not in danger of being sacked, but added that now is not the time to panic.
He said: “We, of course, understand we're not in a good situation, but with everything in life you need to stay calm, keep doing it and keep going.”