The White House was expected to send a letter to House Speaker Nancy Pelosi arguing that Congress cannot conduct an impeachment investigation without first having a vote to authorize it. The letter was expected to say the administration won't cooperate with the probe without that vote.
Trump said the resolution would likely pass, but he predicted it would backfire on Democrats.
"I really believe that they're going to pay a tremendous price at the polls," he said.
When Pelosi announced that the House was initiating the inquiry, she didn't seek the consent of the full chamber, as was done for impeachment investigations into former Presidents Richard Nixon and Bill Clinton.
But it is underway, and with a rapidly escalating pace. Late Thursday, House investigators released a cache of text messages that showed top U.S. diplomats encouraging Ukraine's newly elected president to conduct an investigation linked to Joe Biden's family in return for a high-profile visit with Trump in Washington.
The release followed a 10-hour interview with one of the diplomats, Kurt Volker, who stepped down as special envoy to Ukraine after the impeachment inquiry had begun.
Trump repeated on Friday that he was pressing Ukraine to investigate corruption, not trying to undermine Biden, who could be his 2020 presidential election opponent.
As Republicans search for a response to the investigation, the absence of a procedural vote to begin the probe has been a main attack line against Democrats.
Pelosi swatted the need for such a vote back as unnecessary, saying the House is well within its rules to pursue the inquiry without it.
"The existing rules of the House provide House Committees with full authority to conduct investigations for all matters under their jurisdiction, including impeachment investigations," Pelosi wrote Thursday in a letter to House GOP Leader Kevin McCarthy after he, too, pressed for a floor vote.
Pelosi has sought to avoid a vote on the impeachment probe for the same reason she resisted, for months, liberal calls to try to remove the president: It would force moderate House Democrats to make a politically risky vote.
The White House, meanwhile, is trying to force the question on Democrats, as it seeks to raise the political cost for their impeachment investigation and to animate the president's supporters ahead of the 2020 election.
Trump allies have suggested that without a formal vote, the House is merely conducting standard oversight, entitling lawmakers to a lesser level of disclosure from the administration. The Justice Department raised similar arguments last month, though that was before Pelosi announced the impeachment investigation.
Two days after telling reporters, "Well, I always cooperate," Trump struck a different note on cooperating with the House probe. "I don't know," he said. "That's up to the lawyers."
Democrats have warned that the Trump administration's obstruction of the investigation is itself a potentially impeachable office. The administration was expected to miss various deadlines Friday to comply with House investigators' requests for documents.
There's no clear-cut procedure in the Constitution for initiating an impeachment inquiry, leaving many questions about possible presidential obstruction untested in court, said Allan Lichtman, a history professor at American University.
"There's no specification in the Constitution in what does and does not constitute a more formal impeachment inquiry or investigation," he said.
Rudy Giuliani, the president's personal attorney, dismissed the entire premise of the impeachment inquiry, which is centered on Trump asking Ukraine to investigate his possible political rival, Democratic former Vice President Biden.
"The president was not tasking Ukraine to investigate a political opponent," Giuliani told The Associated Press on Thursday. "He wanted an investigation into a seriously conflicted former vice president of the United States who damaged the reputation of the United States in Ukraine."
Democrats have sought to use their declared impeachment investigation to bolster their case to access all sorts of documents from the administration, most recently secret grand jury information that underpinned special counsel Robert Mueller's report on Russian interference in the 2016 election. They have also threatened to use the administration's refusal to turn over documents and make witnesses available to potentially form an article of impeachment over obstruction of the congressional inquiry.
Where courts have generally required congressional oversight requests to demonstrate a legitimate legislative purpose, impeachment requests could be wide-ranging.
It is unclear if Democrats would wade into a lengthy legal fight with the administration over documents and testimony — or if they would just move straight to considering articles of impeachment.