House Democrats have turned to the courts at an unprecedented tempo in their clashes with Mr. Trump, and they went into court on Tuesday to file yet another case — this one challenging the administration’s defiance of a subpoena for documents about its attempt to add a citizenship question to the 2020 census. But they are also growing disillusioned with the courts as a solution.

The Democrat leading the investigation into the Ukraine affair, Representative Adam B. Schiff of California, has made clear that lawmakers will move forward with weighing articles of impeachment rather than getting bogged down in courts. He used another sports metaphor, the tactic of boxers who lean against the ring ropes and trick their opponents into exhausting themselves by ineffectively pummeling them.

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“We are not willing to go the months and months and months of rope-a-dope in the courts, which the administration would love to do,” Mr. Schiff said on NBC’s “Meet the Press” on Sunday, explaining that he and his colleagues view their investigation as urgent because Mr. Trump has solicited foreign interference in the 2020 election. Because of that, Mr. Schiff said, they will not wait even for witness testimony and documents they would like to obtain.

Indeed, in another major court development on Monday that got far less attention than the McGahn lawsuit, the Supreme Court blocked an appeals court ruling that the House can subpoena Mr. Trump’s financial records while the justices consider whether to hear the case — alongside a similar case involving the Manhattan district attorney’s push for such records.

Both cases generated headlines when district court judges and then appeals courts ruled against the president. But if the Supreme Court does take the appeals, justices may issue no final judgment until the court’s term ends in seven months.

Mr. Trump suggested he instead had a more principled motive than running out the clock on Tuesday, claiming in a series of tweets that he would be happy to let his current and former aides tell Congress what they know, and insisting that he is only blocking them from talking to ensure that “future Presidents should in no way be compromised.”

“I am fighting for future Presidents and the Office of the President,” Mr. Trump said. “Other than that, I would actually like people to testify.”