Exxon Mobil Corp. and other energy companies have joined President Donald Trump in expressing concerns over a bill to toughen sanctions on Russia, arguing that it could shut down oil and gas projects around the world that involve Russian partners.

The pushback from energy companies such as Exxon and Chevron Corp.—and other industries—threatens to complicate House passage of the legislation, aimed partly at punishing Russia for what U.S. officials describe as interference in last year’s U.S. election. The bill breezed through...