WALL STREET JOURNAL, KRISTINA PETERSON and KATE DAVIDSON

House Speaker Nancy Pelosi and Treasury Secretary Steven Mnuchin continued to haggle over policies including testing strategy in their negotiations Wednesday over another coronavirus relief package, despite closing some distance over how much money to spend.

The two are expected to speak again Thursday. The $1.88 trillion proposal from the White House, formally submitted over the weekend, sparked some calls from Democrats this week to accept the administration’s newly boosted offer. But Mrs. Pelosi (D., Calif.) continued to highlight her opposition to policy components of the White House proposal, though its spending level moved closer to the $2.2 trillion legislation passed by the House earlier this month.

Mr. Mnuchin said that although he and Mrs. Pelosi were making progress on certain issues, disagreements remained not only on the size of the bill, but on policy measures.

“There’s a lot of detail other than just the top-line number that we’re exchanging paper [on]. We continue to make progress on certain issues, on certain issues we continue to be far apart,” Mr. Mnuchin said Wednesday in an online discussion hosted by the Milken Institute. “At this point, getting something done before the election and executing on that will be difficult.”

Mrs. Pelosi and Mr. Mnuchin spoke for roughly an hour Wednesday morning and had a productive discussion clarifying details of proposals they exchanged over the weekend, Drew Hammill, Mrs. Pelosi’s spokesman, said on Twitter.

Mrs. Pelosi continued to press for $75 billion for testing and tracing of the virus, as well as a national plan for implementing tests.

“The White House lacks an understanding of the need for a national strategic testing plan,” Mr. Hammill said. The White House proposal included about $45 billion in new funding for testing and tracing, Mrs. Pelosi said over the weekend.

The Trump administration has sent funds to states to spend at their discretion on testing and contact tracing. It has helped states source supplies such as swabs and chemicals used to process tests, and more recently has placed large orders of rapid tests to dispatch to nursing homes, assisted-living facilities and state governors to distribute at their discretion.

“We’re very pleased with the rate of progress on vaccines and testing. Unlike previous financial crises, the end will be in sight on this shortly,” Mr. Mnuchin said.

Administration officials have said states are best equipped to determine what they need and work with the federal government on their respective testing plans. Democrats have said that approach has been a failure that allowed the virus to percolate rather than mitigate its spread and called instead for a centralized, federally led approach to testing with consistent data collection that they say would more efficiently route resources to where they are needed.

“We cannot safely reopen schools, the economy and our communities until we crush the virus with the science-based national plan for testing, tracing, treatment and isolation,” Mrs. Pelosi said in a letter to House Democrats over the weekend.

While Democrats broadly back a national testing plan, some don’t think its absence is worth derailing a relief package likely to contain another round of direct checks to Americans, federal unemployment insurance and funding for schools, which have bipartisan support, although details remain unresolved.

“The people in my district who come up to me and say they don’t know how they’re going to pay their rent…It’s not going to matter very much to them if they lose their business now if we continue arguing over the next three or four months over a national testing strategy,” said Rep. Abigail Spanberger (D., Va.).

On Tuesday, Rep. Ro Khanna (D., Calif.) called on Mrs. Pelosi to accept the White House offer rather than risk extending the impasse until the next administration.

“People in need can’t wait until February. 1.8 trillion is significant & more than twice Obama stimulus,” Mr. Khanna said on Twitter. “Make a deal & put the ball in McConnell court,” she added, referring to Senate Majority Leader Mitch McConnell (R., Ky.).

It isn’t clear that even if Mrs. Pelosi and Mr. Mnuchin were to reach an agreement, it would be able to pass the GOP-controlled Senate. Mr. Mnuchin and White House chief of staff Mark Meadows faced resistance from Senate Republicans on a call over the weekend. Republicans were critical of the proposal’s overall spending level and provisions including an expansion of the Affordable Care Act subsidies for people who lost employer-sponsored health care during the pandemic.

Sen. McConnell said Tuesday that the Senate would vote on a narrow coronavirus relief bill next week. Mr. McConnell said it would include new funding for the Paycheck Protection Program, a $670 billion federal initiative that provides forgivable government-backed loans to small businesses grappling with the economic fallout of the pandemic. Authorization for PPP expired in August with roughly $130 billion of the funds left over.

Mrs. Pelosi has said that passing a narrow bill would be a missed opportunity and exclude priorities important to Democrats, including state and local funding.

In an interview with Fox Business Network Wednesday, Mr. Mnuchin reiterated that about $300 billion in unspent funds that Congress authorized in the March Cares Act could be repurposed immediately for additional aid to small businesses and airlines. That includes funding left over from the Payroll Protection Program, and money provided to the Treasury Department to support Federal Reserve lending programs. Those measures are supported by lawmakers on both sides of the aisle.

“I suspect that they don’t want to give the president a win three weeks before the election,” Mr. Mnuchin said of Democrats, “but they are holding up relief to hardworking Americans.”