Paul D. Kamenar, a lawyer for Mr. Miller, said they would probably appeal, although they had not yet decided whether to ask the full appeals court to review the panel’s decision or whether to take the matter directly to the Supreme Court.

“We are disappointed with the decision and will be considering future legal action whether before the full court of appeals or the Supreme Court,” he said in a phone interview.

Essentially, Mr. Miller’s legal team had argued that Mr. Mueller wielded too much power to qualify as an “inferior officer” under the Constitution, the type who can be appointed without Senate confirmation. Mr. Mueller was appointed in that manner when the deputy attorney general, Rod J. Rosenstein, named him as special counsel in May 2017. If Mr. Mueller was improperly appointed, they said, the subpoena was invalid.

But at a hearing over the dispute before the appeals court panel in November, Michael R. Dreeben, one of the lawyers working for Mr. Mueller, argued that Mr. Mueller was not “off in a free-floating environment,” but rather was subject to the supervision and control of the attorney general. As a result, he maintained, there was no need for Mr. Mueller to have been confirmed by the Senate. The appeals court agreed.

Mr. Miller’s attempt to challenge Mr. Mueller’s authority was similar to an earlier challenge brought by Paul Manafort, President Trump’s former campaign chairman, who was prosecuted by Mr. Mueller’s office. Mr. Manafort’s gambit also failed, and he was convicted after a trial on bank and tax fraud charges in the Eastern District of Virginia and then pleaded guilty to separate conspiracy charges in the District of Columbia.