WASHINGTON—A government watchdog recommended that top White House adviser Kellyanne Conway be removed from her post, citing repeated alleged violations of a rule that prohibits political activity by executive-branch staff.

In a statement Thursday, the Office of Special Counsel—which isn’t affiliated with former special counsel Robert Mueller’s Russia investigation—described Ms. Conway as a “repeat offender” of the Hatch Act for attacking Democratic presidential candidates while speaking in her official capacity as a White...