WASHINGTON—The Senate Judiciary Committee plans to open an investigation into the circumstances surrounding President Donald Trump’s removal of James Comey as FBI director, a probe that could examine the thorny question of whether Mr. Trump improperly interfered in a continuing investigation by doing so.

“The Judiciary Committee has an obligation to fully investigate any alleged improper partisan interference in law enforcement investigations,” said Sen. Chuck Grassley, an Iowa Republican and committee chairman, said in a...