WASHINGTON—The U.S. has complained to senior Russian officials about an Iranian general’s recent trip to Moscow, State Department spokesman Mark Toner said.

The U.S. will also raise its concerns at the United Nations and pursue the matter through the Security Council. U.S. officials said they believe Maj. Gen. Qasem Soleimani, head of Iran’s elite military Qods Force, went to Russia in late July to meet officials there in violation of U.N. travel ban.

“We’ve raised this travel (concern) with senior Russian foreign ministry officials and we’re going to raise it and address it further in New York,” Mr. Toner said Wednesday.

The complaint, and Gen. Soleimani’s trip, come at a sensitive time for the Obama administration. It is working to sell skeptical lawmakers on a nuclear agreement it reached with Iran and several other countries that it has said will prevent Iran from developing a nuclear weapon in exchange for relief from numerous sanctions. Many critics of the White House’s agreement have said Iranian officials cannot be trusted, and Gen. Soleimani’s trip to Russia despite his travel ban could further inflame critics.

The State Department is now pushing for an investigation into Gen. Soleimani’s travels. The U.S. hasn’t received any responses from Russia to its concerns, Mr. Toner said. Iranian state media reported that Gen. Soleimani discussed weapons sales with Russian officials during his visit.