“Border security and immigration issues are important, and we should deal with them, but we can deal with several important issues at the same time,” said Senator Susan Collins, Republican of Maine. “We can’t put the rest of the business of government on hold while these issues, important though they are, are resolved.”

For House Democrats, the return to normal governance provides a window to reclaim some of the attention the shutdown drained from the initial rollout of their legislative agenda. House Democratic leaders had insisted when they took control early this month that the shutdown would not affect their carefully choreographed agenda rollout. But it did.

Speaker Nancy Pelosi of California and her allies plan to hold a ceremony on Wednesday to formally reintroduce the Paycheck Fairness Act, a measure intended to equalize pay between men and women that Democrats have tried to enact for 20 years. Like other bills under consideration in the House, it is unlikely to be taken up by the Republican-controlled Senate, but it is a key piece of Democrats’ messaging to voters ahead of the 2020 elections.

Ms. Pelosi and Mr. Trump also are likely to renew discussion in the coming days about a date to reschedule the president’s State of the Union address, which was previously scheduled for Tuesday and became a casualty of the shutdown. Last week, Ms. Pelosi rescinded her invitation for the president to address Congress while the government was still shuttered, and has said subsequently that the two sides will work to find an agreeable date in the future to allow time to prepare.

On the House floor, Democrats will bring up a vote on legislation that would give nonmilitary federal workers a pay raise in line with the pay increase members of the military have received. The legislation, if it became law, would override an executive order issued by Mr. Trump a week into the shutdown that froze civilian federal pay.

The House Armed Services Committee will hold a hearing on Tuesday about the cost and rationale behind the deployment by Mr. Trump of thousands of active duty troops to the southern border. Representative Adam Smith of Washington, the committee’s chairman, said in a statement that “there has not been adequate justification for the use of our military personnel in this way.”