WASHINGTON—A push to complete a roughly $1 trillion infrastructure agreement hit a series of hurdles Monday, as aides squabbled over funding for water infrastructure and how to apply a requirement that federal contractors pay their employees a locally prevailing wage, among other issues.

Lawmakers had previously set Monday as a target for closing out their talks and beginning floor consideration of the emerging agreement, though that timeline seemed to slip as the two sides sniped at each other.

Democrats sent Republicans an offer Sunday night to try to resolve the remaining issues, which also include transit funding and repurposing Covid-19 aid for infrastructure, according to a Democratic aide familiar with the talks.

Republican aides panned the offer on Monday, accusing Democrats of backtracking on parts of the agreement.

The offer “was discouraging since it attempts to reopen numerous issues the bipartisan group had already agreed to,” a Republican aide familiar with the talks said.

Other aides familiar with the talks said negotiators were also still trying to figure out how to cover the cost of the spending with a mix of new revenue and savings measures.

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

The core group of 10 senators who have spent weeks negotiating the details of the agreement—which they first unveiled last month at the White House with President Biden—were set to meet again Monday evening. White House press secretary Jen Psaki cast the flare-up of disagreements in the talks as part of the process toward achieving an agreement.

“We always know that there is some wrangling at the end of a process,” Ms. Psaki said.

At stake is the fate of both the infrastructure plan—of which about $600 billion would be spending above projected federal levels if current programs continued—and President Biden’s broader goals on Capitol Hill. Top Democrats see the infrastructure bill as the first plank of approving much of Mr. Biden’s $4 trillion agenda and are hoping to follow it with a $3.5 trillion bill that they plan to advance through a budget process that avoids the 60-vote threshold necessary for most legislation in the Senate.

Senate Majority Leader Chuck Schumer (D., N.Y.) has said he wants to approve the infrastructure deal and an outline of the $3.5 trillion bill before the Senate is set to leave town next month, creating a tight timetable to make progress on a variety of policy questions. An initial procedural vote on the infrastructure plan failed last week, as Republicans said they were unwilling to support the effort until they had completed the deal’s details.

The dispute around water infrastructure revolved around how to incorporate a $36-billion, Senate-passed water reauthorization bill into the deal, according to aides in both parties. Republicans charged Democrats with suddenly demanding more funds for water infrastructure, while Democrats said Republicans were abandoning a previous understanding on the issue.

Negotiations on transit funding continued Monday, as Democrats sought a greater share of funding for public-transit systems than Republicans.