“President Biden is now a wartime commander in chief waging four wars at once,” Mr. Bash said on Saturday. “An economic war, an information war, likely a cyber war, and an unprecedented indirect military war against Putin. And so far, Putin has been unable to achieve a single one of his objectives.”

Several of the administration’s most ardent supporters in the foreign policy world quickly chided the president for seeming to seek Mr. Putin’s removal. Richard Haass, the president of the Council on Foreign Relations, called it a “bad lapse in discipline that runs risk of extending the scope and duration of the war.”

While American officials still insist their goal is not regime change in Moscow, even the president’s top national security advisers have made clear they want Mr. Putin to emerge strategically weakened.

“At the end of the day, the Russian people are going to ask the more fundamental question of why this happened and how this happened,” Jake Sullivan, the president’s national security adviser, told reporters on Air Force One on Friday, before the president’s speech. “And we believe that, at the end of the day, they will be able to connect the dots.”