WASHINGTON—The Republican chairman of the Senate Judiciary Committee said Wednesday his panel would take up a bill to protect special counsel Robert Mueller from being dismissed without cause, the first major congressional action designed to protect the integrity of the criminal investigation into Russian activity during the 2016 election.

A spokesman for Sen. Chuck Grassley, an Iowa Republican, said that he would put a bipartisan bill that would prevent Mr. Mueller from being dismissed without cause on the committee’s agenda....