The resolution, which would give Mr. Trump a 30-day deadline to come to Congress for authorization for military action in Iran, would still need to be passed by the House. And it would be unlikely to overcome a veto from the president.

Mr. Kaine introduced the measure, which invokes the War Powers Act of 1973, as a privileged joint resolution, which allows him to force a vote on the measure and win over the support of a simple majority of senators.