Left without the power to filibuster, Democrats are settling on another strategy to derail President-elect Donald Trump’s nominees: Demand more information.

It’s the same tactic that helped Democrats sink some nominees the last time they were in the minority, during the George W. Bush administration. But they may have dented their own cause by powering through President Obama’s picks and drawing a road map that Republicans will now employ to confirm Mr. Trump’s nominations.

The first target for the strategy is Rep. Tom Price, the anti-Obamacare crusader whom Mr. Trump tapped to run the Department of Health and Human Services.

Senate Democrats demanded Thursday that Mr. Price’s nomination be put on hold until after Congress investigates his stock trades involving health care companies, even as he worked on health care policy on Capitol Hill.

An investigation could take months, but Democrats said it is needed if lawmakers are to have confidence in their votes.

“This is serious stuff,” said Senate Minority Leader Charles E. Schumer, New York Democrat, though he conceded that he did not have any evidence of wrongdoing.

Mr. Schumer said other nominees are also moving too slowly to get information to senators ahead of a slate of hearings next week, and he threatened to delay the process unless the nominees produce more documents.

“For positions of such influence in our government, it’s the responsibility of the Senate to guarantee that we have all the information we need on each nominee in a timely fashion,” he said. “But truth be told, the slate of nominations selected by President-elect Trump has made this process, standard for nominees of presidents of both parties, immensely difficult.”

Fights over the extent of information to which Congress is entitled have snared other nominees. Democrats derailed Mr. Bush’s nomination of lawyer Miguel Estrada to the powerful U.S. Circuit Court of Appeals for the District of Columbia after accusing him of withholding information.

But Democrats also have pushed through nominees under investigation. In 2013, they powered through the nomination of Alejandro Mayorkas to be deputy secretary at the Department of Homeland Security, even as the inspector general was investigating whether he inserted himself into immigration cases to assist wealthy and politically connected Democrats.

In Mr. Price’s case, the Trump transition team said he took the same actions as many other members of Congress.

“The same questions being raised today by Sen. Schumer about Dr. Price should be directed to Sens. [Thomas R.] Carper, [Mark R.] Warner and [Sheldon] Whitehouse, who own and have traded hundreds of thousands of dollars in pharmaceutical and health insurance company stocks,” said Phil Blando, a transition spokesman. “The reality is that Dr. Price’s 20-year career as an orthopedic surgeon and a fiscal conservative make him uniquely qualified to lead HHS.”

Mr. Price has provided all the required documents to the Senate Finance Committee, which has partial oversight authority for the Department of Health and Human Services.

Committees typically require a set of four documents from Cabinet nominees: a questionnaire that details personal information and qualifications, a financial disclosure, a letter from the U.S. Office of Government Ethics certifying agreement to the ethics code and an FBI background check.

For some positions, such Environmental Protection Agency administrator, an FBI check is not required.

Most of Mr. Trump’s nominees whom Democrats have targeted have provided either all of the documents or are waiting for paperwork from the ethics office or the FBI.

Secretary of State-designate Rex Tillerson, the Exxon Mobil CEO who tops the Democrats’ list of nominees to either delay or defeat, has delivered all of the paperwork except the FBI check to the Senate Foreign Relations Committee.

That has not fully satisfied the demand for paperwork from the committee’s top Democrat, Sen. Benjamin L. Cardin of Maryland. He has requested three years of tax returns from Mr. Tillerson, although that is not a usual requirement for the post. Committee Chairman Bob Corker, Tennessee Republican, has said that he will not enforce the request.

The committee’s Democrats also charge that the Trump transition team did not adequately vet the nominee, so Mr. Tillerson must face more intense scrutiny on Capitol Hill.

Republicans on the committee insist that Mr. Tillerson is meeting or exceeding the same standards applied to Hillary Clinton for her 2009 confirmation as secretary of state.

Mr. Tillerson submitted a completed questionnaire to the committee three days after Mr. Trump announced his nomination. It took Mrs. Clinton 17 days after her nomination to complete her questionnaire.

Both submitted financial disclosures eight days before their hearings. Mr. Tillerson will face the committee Wednesday.

The committee expects the FBI report on Mr. Tillerson in the coming days. The FBI check on Mrs. Clinton did not arrive until two days after her confirmation hearing on Jan. 13, 2009.

Senate Democrats balked at the comparison to Mrs. Clinton, who had been first lady, a U.S. senator and a presidential candidate before her nomination as secretary of state.

“The paper trail was substantial, so it’s not apples to apples. It not a reasonable comparison,” said Cardin spokesman Sean Bartlett.

Nominees who have provided all the required paperwork include billionaire investor Steve Mnuchin, nominee for Treasury secretary, and Oklahoma Attorney General Scott Pruitt, nominee to run the EPA.

Mr. Mnuchin and Mr. Price also provided the last three years of tax returns as part of their required documents for the Senate Finance Committee.

Others who have completed the questionnaire and financial disclosure include Attorney General-designate Jeff Sessions, Education Secretary-designate Betsy DeVos and CIA Director-designate Mike Pompeo,

“Of course, like all nominees, there will be additional submissions” from Mr. Sessions, said Beth Levine, a spokeswoman for Senate Judiciary Committee Chairman Chuck Grassley, Iowa Republican.

She noted that Eric H. Holder Jr. submitted hundreds of documents more than a year after he was confirmed as attorney general in 2009.

“Any notion that Sen. Sessions hasn’t made a good-faith effort to supply the committee with responsive material is ridiculous,” said Ms. Levine.

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