President Trump dug in Wednesday against House Democrats’ expanding investigations of his administration and business empire, accusing liberals of trying to win the 2020 presidential election with a blizzard of subpoenas.

Declaring that his cooperation with special counsel Robert Mueller’s probe showed that he’s “the most transparent president and administration in the history of our country,” Mr. Trump warned that he’ll take Democrats to the Supreme Court if they persist with demands for more documents and testimony aimed at possible impeachment.

“I thought after two years, we’d be finished with it,” the president told reporters at the White House. “I say it’s enough. We’re fighting all the subpoenas.”

House Judiciary Committee Chairman Jerrold Nadler of New York, who reportedly has suggested fining administration officials who refuse to comply with subpoenas, called the president’s resistance “one more act of obstruction.”

House Oversight and Government Reform Chairman Elijah E. Cummings of Maryland, already waging a subpoena feud with the Justice Department, said the administration is engaged in a “massive, unprecedented, and growing pattern of obstruction.”

So far this week the Treasury Department has rebuffed a deadline to turn over years of Mr. Trump’s tax returns, and the president went to court to block an effort by House Democrats to obtain eight years of financial statements from an accounting firm for the president and his business empire.

Meanwhile, Judiciary Committee Democrats issued a subpoena for former White House Counsel Don McGahn, saying they want to probe his actions in defending the president in Mr. Mueller’s Russia investigation. That investigation found no collusion between the Trump campaign in 2016 and Russian meddling in the election.

Mr. Trump said Democrats’ demands for more investigations are motivated by politics.

“The Democrats are trying to win 2020,” the president said. “They’re not going to win with the people that I see. And they’re not gonna win against me. The only way they can maybe luck out, and I don’t think that’s gonna happen … the only way they can luck out is by constantly going after me on nonsense.”

Mr. Trump, while still complaining that the Mueller report was a “witch hunt,” says it was thorough enough that there’s no need for Democrats to do their own probes.

In particular, he said he figures Mr. Mueller already pored over his finances and checked his taxes.

The president said the subpoena for Mr. McGahn is “ridiculous” and said he’s run the most transparent administration in history, noting that he turned over millions of pages of documents and didn’t assert executive privilege on any of the special counsel’s requests for information.

White House Counselor Kellyanne Conway said the president could assert executive privilege to block testimony by current and former White House officials including Mr. McGahn.

“Executive privilege is on the table,” she told reporters. “That’s his right … But we should note that there’s been a great deal of executive cooperation and compliance.”

Mr. Nadler, though, said the time “has long since passed” for the president to assert executive privilege to prevent Mr. McGahn from testifying to Congress.

The Mueller report said the president told Mr. McGahn to order Deputy Attorney General Rod Rosenstein to fire Mr. Mueller in 2017, but Mr. McGahn failed to deliver that message. Some Democrats argue the directive from Mr. Trump could constitute obstruction of justice and spur impeachment proceedings.

The president said he would take House Democrats to the Supreme Court if they try to impeach him, saying that Mr. Mueller “didn’t lay a glove on me.”

“If the partisan Dems ever tried to Impeach, I would first head to the U.S. Supreme Court,” the president said on Twitter, then breaking into all-capital letters to declare that “I did nothing wrong.”

The House holds the power to carry out impeachment proceedings. Article One, Section Three of the Constitution grants the Senate “the sole power to try all impeachments.” The chief justice of the Supreme Court, currently John G. Roberts Jr., presides over impeachment proceedings.

But the high court cannot overturn an impeachment vote, and the justices have generally taken a hands-off approach to the procedure. The court ruled in 1993 in the impeachment of federal Judge Walter Nixon that questions about the Senate’s conduct of an impeachment trial constituted a political question, and therefore couldn’t be resolved in the courts.

Chapman University law professor Tom Campbell, a former five-term House Republican who served on the Judiciary Committee, said there is no role for the Supreme Court to rule on what the House “considers relevant to its consideration of impeachment.”

“Federal courts may rule on subpoena requests from congressional committees, including resolving issues of executive privilege and attorney-client privilege,” Mr. Campbell said. “However, if the president disobeys an order to provide information subpoenaed by a congressional committee, the Congress lacks any enforcement power — except to consider the president’s refusal to supply that information as grounds for impeachment.”

But courts could be asked to referee other disputes, including the demands for documents and testimony that House Democrats have fired at the administration.

The Justice Department on Tuesday told Mr. Cummings it won’t allow John Gore, a key official involved in adding a citizenship question to the 2020 census, to testify Thursday in response to a subpoena.

Attorney General William P. Barr said Mr. Cummings has refused to let a department lawyer be present to assert the administration’s rights, and without that, there can be no deposition.

Mr. Cummings said his investigators will show up anyway, and he warned Mr. Gore to “think very carefully” before sticking with the president on this fight.

“Both President Trump and Attorney General Barr are now openly ordering federal employees to ignore congressional subpoenas and simply not show up — without any assertion of a valid legal privilege,” the congressman said.

Mr. Trump, who is scheduled to meet next week with House Speaker Nancy Pelosi about a possible infrastructure package, said House lawmakers should be focused on the nation’s needs.

“Get back to infrastructure,” he said. “Get back to cutting taxes. Get back to lowering drug prices.”

Before the midterm elections, Mr. Trump told The Washington Times in an interview that he would take Mrs. Pelosi and House Democrats to the Supreme Court if she tried to weaponize her post by using subpoena power as leverage to gain his support for legislation. Now he’s entering their meeting facing numerous subpoenas, although Mrs. Pelosi has been tamping down calls for impeachment.

Mr. Trump’s 2016 opponent, Hillary Clinton, weighed in with an op-ed in The Washington Post on Wednesday saying Congress should be open to impeachment, but shouldn’t rush to it. She urged them to avoid the mistakes she said Republicans made in 1998 when they impeached her husband, former President Bill Clinton, but failed to muster the votes in the Senate to oust him.

Mr. Trump said Congress should be investigating Mrs. Clinton and the FBI officials who started the counterintelligence probe of his campaign in 2016.

“If you want to litigate, go after the DNC, crooked Hillary, the dirty cops, all of these things,” the president told reporters. “That was a rigged system and I’m breaking down — I am breaking down the swamp. If you look at what’s happening, they’re getting caught, they’re getting fired. I am draining the swamp.”

• Stephen Dinan and Bailey Vogt contributed to this report.

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