By Kate Davidson

WASHINGTON -- The Trump administration is weighing a second round of stimulus payments for Americans as part of an economic-relief package Congress is likely to consider next month, Treasury Secretary Steven Mnuchin said Thursday.

Mr. Mnuchin said he had discussed with President Trump the idea of additional stimulus payments, though no decision had been made yet on whether to advocate for them in the next bill.

"It's something that we're very seriously considering," he told reporters during an online question-and-answer session Thursday.

Congress provided an initial round of onetime payments of $1,200 for most adults and $500 for children under age 17 as part of the Cares Act enacted in March. The Internal Revenue Service said it has distributed payments to all eligible Americans for whom it has sufficient information, totaling $267 billion.

That money helped households fill holes in their budgets and propped up consumer spending as the economy struggled in April and May. The House of Representatives voted for a second round of payments last month as part of its $3.5 trillion economic-relief package. Those payments would be larger, offering $1,200 each for up to three children instead of $500. The Democratic proposal also covers groups left out of the first round of payments, such as college students, adult dependents and households that include undocumented immigrants.

Mr. Mnuchin also said it is extremely unlikely that parts of the U.S. economy will need to shut down again, despite a surge in coronavirus cases in some parts of the country. Mr. Mnuchin said he expected officials will make sufficient medical progress between now and the end of the year, including more widespread testing and effective viral treatments, that will support safe reopening of the economy.

"Could there be some rare, extreme scenario that occurs that, based upon medical advice, the president does" close down the economy, he said. "I think that's extremely unlikely."

The Trump administration and lawmakers are weighing how much additional support to provide in the months ahead, a decision that depends on how quickly the U.S. economy snaps back this summer. Recent data, including an encouraging May jobs report and steadily falling initial jobless claims, suggest an upturn has already started, though many economists expect a long, slow recovery.

Mr. Mnuchin emphasized that economic relief in the next measure should be targeted at industries and businesses that face a more difficult recovery. But his comments suggest the administration is sympathetic to the idea of more aid for struggling workers and families, despite concerns from some Republicans over additional spending that would push deficits even higher.

Congress has already provided about $3.3 trillion of spending and tax breaks, including the stimulus payments, emergency small-business loans and enhanced unemployment benefits.

Mr. Mnuchin said it was premature to speculate on the overall size of the next relief package.

"Before we rush back and spend more money, whether that's a trillion dollars or whether that's more, we want to make sure we're careful in knowing how much more we need to spend," Mr. Mnuchin said.

One potential flashpoint in negotiations over the next bill is the extra $600 in weekly jobless benefits set to expire at the end of July. The additional payments were aimed at covering workers' lost paychecks while businesses across the country were closed. Many workers have made more money from benefits than they did on the job, and Republicans have warned it might discourage them from returning to work.

"If we continue enhanced unemployment, we're obviously going to need to fix that," Mr. Mnuchin said, "so that we don't end up in a situation where workers are paid more on unemployment than they are to go back to work."

"I'm very focused on how do we create the right motivation," he added.

Mr. Mnuchin said the use of retention tax credits enjoyed broad support in the Cares Act. "I could see expanding that in some type of rehire tax credit that is perhaps even more impactful," he said.

Congress included the retention credit in the March legislation and it is effectively a government wage subsidy. It covers up to 50% of wages and benefits up to $10,000 for employers suffering from closures or significant revenue declines. Smaller companies get the break for all workers; larger firms get it for paying people not to work.

The House legislation includes a $194 billion expansion of the credit, and it has bipartisan support. That version would cover 80% of costs up to $45,000 and add a credit for fixed expenses such as rent.

Write to Kate Davidson at kate.davidson@wsj.com