The 22-year-old will see a specialist to find out the extent of the injury, but recovery time is usually six weeks.
Underhill was part of England's 2018 Six Nations side and featured in three of the autumn internationals.
England start their 2019 Six Nations campaign with a visit to defending champions Ireland on 2 February.
Bath will also lose lock Matt Garvey, who fractured his ankle in the same game, for around three months.
Underhill, who has nine caps for his country, is not the only international player in recovery at Bath, with England winger Joe Cokanasiga out with a medial knee ligament injury. The 21-year-old is not expected back until mid-January.
Backs Anthony Watson and Jonathan Joseph are also out with Achilles and mid-foot ligament issues until "early 2019", while Wales back-row Taulupe Faletau is expected to return soon from a fractured forearm.