Many Democrats involved in the inquiry already believe they have collected enough to impeach him for abusing his power by enlisting a foreign government to smear his political rivals. But to persuade the public — and the necessary number of Republican senators — that the president should be convicted and removed from office, they may need additional proof tying him directly to certain elements of the alleged wrongdoing. They could potentially unearth stronger evidence by turning to the courts, but that could also stall the case for months and risk losing public support, much as some Democrats believe happened in the Russia inquiry.

“As in many investigations, you get to a point where you have to decide how much is enough and whether the incremental value of the additional juice is worth the squeeze,” said Ross H. Garber, a lawyer who is one of the nation’s leading experts on impeachment. “ If anything, they may be surprised by how much cooperation they have gotten from witnesses already, notwithstanding the position of the executive branch.”

For now, Democrats have not yet exhausted testimony from officials who appear willing to cooperate and have at least peripheral knowledge of the case. At least two more White House officials are scheduled to testify this week, and are expected to confirm key events . Other officials from the State and Defense Departments involved in Ukraine policy are set to appear, as well.

Democrats believe a reconstructed transcript released by the White House of a July phone call between Mr. Trump and President Volodymyr Zelensky of Ukraine also significantly bolsters their case. In the call, Mr. Trump pressed Mr. Zelensky to investigate former Vice President Joseph R. Biden Jr. and unproven theories about Democratic collusion with Ukraine during the 2016 election.

“We have a tremendous table of evidence before us that fills in all of the principal, material questions that were raised by the whistle-blower,” said Representative Jamie Raskin, Democrat of Maryland, referring to an anonymous C.I.A. whistle-blower whose complaint about Mr. Trump’s actions toward Ukraine helped prompt the impeachment inquiry.