WASHINGTON—A Senate panel on Thursday overwhelmingly approved a bill that would protect special counsel Robert Mueller from being fired without cause, in a bipartisan show of support for the ongoing investigation into Russian interference in the 2016 election.

Serious political and legal hurdles stand in the way of it becoming law. The bill, authored by two Democrats and two Republicans, would enshrine into law a Justice Department guideline that a special counsel can’t be fired without cause. In addition, it would give special...