WASHINGTON—Senators wrapped up the construction of a roughly $1 trillion infrastructure bill, racing to pass it through the chamber by the end of the week and send it to the House, where its fate is intertwined with a $3.5 trillion package of Democratic priorities.

Senate Majority Leader Chuck Schumer (D., N.Y.) kept the Senate open through the weekend in a bid to begin considerations of the bipartisan infrastructure package, $550 billion of which is in addition to expected future federal investments in roads, bridges, broadband, and other infrastructure projects. But lawmakers took much of the weekend to complete the text of the bill, finishing the final text Sunday night. Completion of the text will allow lawmakers to begin the amendment process and move to final passage in the coming days.

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“My hope is we’ll finish the bill by the end of the week,” Sen. Susan Collins (R., Maine), one of the 10 senators who negotiated the core of the agreement with the White House, said on CNN’s “State of the Union.

Mr. Schumer said in floor remarks Sunday night that the bill should move through the chamber quickly. “Given how bipartisan the bill is, and how much work has already been put in to get the details right, I believe the Senate can quickly process relevant amendments and pass this bill in a matter of days,” he said.

The text of the bill will translate into law the agreement lawmakers struck days ago, which provides $110 billion to bridges and roads, $39 billion to public transit, and $65 billion for expanding access to broadband, among other investments beyond currently projected spending. Repurposed federal funds, money recouped from delaying a Trump-era Medicare rebate rule, and a mix of accounting maneuvers will cover the cost of legislation.

Seventeen Republicans joined all 50 Democrats on the first procedural vote for the package last week, putting it on track to garner more than enough GOP support to clear the Senate’s 60-vote threshold. A lengthy amendment process this week has the potential to chip away—or add—support for the bill.

The legislation will face a more complicated path in the House, which Democrats control narrowly. House Speaker Nancy Pelosi (D., Calif.) has repeatedly said that the chamber won’t take up the infrastructure agreement until the Senate approves a separate antipoverty and climate package that Democrats expect to pass through a budget process called reconciliation without GOP support.

Mrs. Pelosi’s ultimatum—a version of which President Biden walked away from after blowback from Republicans—is aimed at unifying Democrats around both bills. Moderate Democrats have focused on the infrastructure deal, with some saying they are skeptical of elements of the $3.5 trillion plan. Progressive Democrats, in turn, have raised concerns about the infrastructure deal and advocated for the importance of the broader bill.

The $1 trillion bipartisan infrastructure package and the $3.5 trillion Democratic healthcare and antipoverty plan will face obstacles as they make their way through Congress in tandem. WSJ’s Gerald F. Seib explains. Photo illustration: Laura Kammermann The Wall Street Journal Interactive Edition

Republicans have criticized the $3.5 trillion proposal, arguing that its spending could fuel inflation and its tax increases could slow growth.

Rep. Alexandria Ocasio-Cortez (D., N.Y.) said on CNN’s “State of the Union” that at least 10 House Democrats would vote against the infrastructure bill if it isn’t accompanied by the $3.5 trillion legislation, which is also expected to include tax increases on corporations and high-income households.

“We need a reconciliation bill if we want this bipartisan bill to pass,” she said.

It could take weeks, if not months, for the reconciliation legislation to come together. Mr. Schumer has set a target of passing an initial budget blueprint for the $3.5 trillion bill in the coming weeks. Sen. Bernie Sanders (I., Vt.), the chairman of the Senate Budget Committee, said he expects the Senate to take up a budget resolution next week.

Once that resolution passes, though, lawmakers will still need weeks to craft the specifics of the legislation, likely delaying House passage of the $1 trillion infrastructure bill. That time frame will likely frustrate moderate Democrats in the House, who have pushed for the chamber to take up the infrastructure agreement immediately after it passes the Senate.

Sen. Joe Manchin (D., W.Va.), a pivotal centrist, said on CBS’s “Face the Nation” that the two pieces of legislation didn’t need to be considered together.

“I’ve always believed that everything should rise or fall on its own merits,” he said. “I would never ever ever try to advise Speaker Pelosi on how she runs the House. I think she does a marvelous job and she’ll do what she thinks is best for the House, but the bottom line is, we’ve got a bill…it should fall or rise on its own.”

The two legislative efforts represent the bulk of Mr. Biden’s domestic agenda, setting up a hectic fall on Capitol Hill. Lawmakers will also have to raise the government’s borrowing limit in the coming weeks. A two-year suspension of the debt limit expired July 31, meaning the Treasury will turn to emergency measures to conserve cash and keep paying the government’s bills.

How long those measures will last is uncertain, and the Congressional Budget Office recently estimated that the so-called X date could fall some time in October or November. Some top Republicans have said they may oppose raising the debt limit, calling on Democrats to take up the measure along party lines through reconciliation, in a bid that Democrats denounced as irresponsible.

Budget reconciliation may offer Democrats a way to sidestep some partisan gridlock and advance President Biden’s policy objectives. WSJ explains how the process works and why divisions in the party could scuttle the process. Photo Illustration: Carlos Waters/WSJ The Wall Street Journal Interactive Edition

Write to Andrew Duehren at andrew.duehren@wsj.com