The BBC has confirmed that it had complaints over the incident, and according to reports the number is over 100.
Stevens made the remarks after host Alex Jones asked about his new series Gaslit, which is about Watergate.
He replied: “Well, what you have got is a criminal for a leader who is wrapped in a messy war, embroiled in a stupid scandal and surrounded by ambitious idiots and really should resign.”
The Downton Abbey star went on: “Oh no, I’m sorry that’s the intro to Boris Johnson, I’ve just said the intro to Boris Johnson, I’m so sorry.”
The BBC said on its website that it had “received complaints from viewers who were unhappy” about the comments.
Watch: Dan Stevens trolls Boris Johnson
A statement said: “The One Show is a live topical entertainment programme where the hosts are joined by a variety of guests to discuss a broad range of subjects.
"Actor Dan Stevens was a guest on Wednesday 13 April and made comments about Boris Johnson in response to a question from the hosts about his new TV project before going on to talk about the series.
"These comments were taken in jest and the hosts moved the conversation on to the focus of the interview.”
Stevens plays John Dean, a former attorney who served as White House counsel for US President Richard Nixon.
Yahoo has contacted the BBC and broadcasting watchdog Ofcom.