WASHINGTON—Israel has been conducting an aggressive military campaign across Syria against Iran-backed militia groups, an effort that has been encouraged by the White House but aroused the concern of many U.S. military officials.

While the White House has applauded Israel’s assertiveness against Iran inside Syria, the military officials say they fear it could backfire if Iran’s belief that the U.S. is behind many of the strikes prompts Iran-backed groups to attack American troops in Syria or Iraq.

“It’s...