Congressional Democrats are moving toward a vote this week to define more narrowly their impeachment inquiry into President Trump, setting up the next phase of an effort that has intensely divided the party.

The House Judiciary Committee could vote as soon as Wednesday to outline the scope of its formal impeachment investigation, according to multiple reports.

The expanded effort is among the many priorities lawmakers are set to tackle as Congress returns from its summer break to weigh legislative items on gun control, federal spending and potentially trade.

“When you talk about what the Judiciary Committee’s doing, I’m all in favor of transparency. I like it when they have hearings because the more hearings we have, the more information comes out,” Rep. Matt Cartwright, Pennsylvania Democrat, said on Fox News Channel’s “Sunday Morning Futures” program.

House Judiciary Committee Chairman Jerrold Nadler, New York Democrat, announced a subpoena last week demanding that the Department of Homeland Security turn over documents on any potential pardons Mr. Trump had discussed giving officials for carrying out his immigration policies.

The subpoena is part of the committee’s investigation into whether it should recommend formal articles of impeachment, though the panel itself has not yet voted to give him official powers.

House Democrats said in court filings over the summer that their investigations of Mr. Trump were moving toward impeachment proceedings, and it appears they are ready to take more formal steps in that process this week.

Rep. Doug Collins of Georgia, the top Republican on the Judiciary Committee, said Sunday on Fox News that House Democrats are making things up as they go along.

“They just keep digging their hole. They just keep going after things because they hate the president so much,” Mr. Collins said. “If they really want to do this, they have to bring impeachment to the floor. This is simply a show. It is a travesty, and frankly they should be ashamed.”

Congress has other issues it could address as lawmakers return to Capitol Hill.

Gun control

After shootings in Texas and Ohio last month left more than 30 people dead, Democrats urged Senate Majority Leader Mitch McConnell, Kentucky Republican, to call the Senate back early and take up House-passed legislation to tighten background checks on gun purchases.

Mr. McConnell said last week that he is waiting for specific cues from the White House on what Mr. Trump would support as part of a broader package.

“The president needs to step up here and set some guidelines for what he would do,” Sen. Roy Blunt, Missouri Republican, said Sunday on NBC’s “Meet the Press.”

Democrats said that attitude is unacceptable.

“Mitch McConnell has basically deflected to the president,” Sen. Amy Klobuchar of Minnesota, a 2020 Democratic presidential candidate, said on CNN’s “State of the Union.” “Show some leadership. Let these bills come up. The House passed them with some Republican support.”

The White House has indicated it is preparing a package that could include legislation to expedite the death penalty for people found guilty of mass shootings.

Mr. Collins predicted that the House Judiciary Committee will take up gun legislation “that was designed to make people feel better about that they’re actually doing something, but in the end actually won’t do anything.”

Democrats have also pressed for action on legislation that would ban certain kinds of semi-automatic, military-style firearms and high-capacity magazines, as well as “red flag” legislation that lets law enforcement temporarily take guns away from dangerous people.

Senate Judiciary Committee Chairman Lindsey Graham, South Carolina Republican, has expressed optimism about crafting red flag legislation that can win bipartisan support, though gun rights advocates have raised concerns repeatedly about due process issues with such bills.

Spending/appropriations

Congress passed a budget deal earlier this year to boost discretionary spending caps by about $320 billion over the next two years, but lawmakers still need to approve fiscal 2020 spending bills before an Oct. 1 deadline to avert another potential government shutdown.

House Majority Leader Steny H. Hoyer, Maryland Democrat, recently suggested to colleagues that the House would try to pass a shorter-term stopgap spending bill to keep the government running into the new fiscal year in October.

The Democrat-controlled House passed most of its annual spending bills before lawmakers departed for their summer vacation, but any legislation will need support from the Republican-controlled Senate as well as the signature of President Trump.

Trade

Congress could also take up Mr. Trump’s proposed rewrite of the North American Free Trade Agreement (NAFTA).

Mr. Collins said he would like lawmakers to take up the U.S.-Mexico-Canada Agreement, known as the USMCA.

“I think that’s something that we need to happen,” Mr. Collins said.

Vice President Mike Pence has been one of the administration’s chief champions of the agreement, but Democrats have expressed concerns.

House Speaker Nancy Pelosi, California Democrat, spoke by phone about the agreement with Canadian Prime Minister Justin Trudeau last week and expressed concerns about labor standards and other issues, according to the Reuters news agency.

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