Senate Majority Leader Mitch McConnell began to sketch out details of a forthcoming GOP-drafted coronavirus relief package Monday, leaving the door open to another round of direct payments to households as well as more funding for state and local governments.

Speaking during a visit to Flaget Memorial Hospital in Bardstown, Kentucky, McConnell said the next round of checks sent to individuals and families — up to $1,200 per person in the March aid package — may be targeted at a lower-income group this time.

“I think the people who have been hit the hardest are people who make $40,000 a year or less. Many of them work in the hospitality industry,” he said.

The $2 trillion March aid package’s maximum payments began phasing out at 5 percent of adjusted gross income above $75,000 for single filers, and above $150,000 for married couples filing jointly. Payments phased out completely for individuals starting at $99,000 annually, or $198,000 for joint filers.

Another $500 was provided for each child under the age of 17, however, so parents with children could still receive some money if their income wasn’t too high to lose all eligibility.

McConnell didn’t rule out additional aid for state and local governments, a key priority for Democrats and governors of both parties. He emphasized that $150 billion in direct aid was already provided in the March aid package that also contained the tax payments to households, however. That doesn’t include additional indirect aid like money for schools and transportation that flow through state and local authorities.

McConnell also began to outline how liability protections for businesses, schools and others that reopen would work. He said that “unless grossly negligent or intentionally engaged in harmful behavior,” stakeholders would be insulated from lawsuits retroactive to December 2019 and through 2024.

“I’ll be unveiling something, which will be a starting place, in a few weeks. And we’ll then be dealing with the administration and the Democrats and all the rest,” McConnell said. “I can’t comfortably predict we are going to come together and pass it unanimously like we did a few months ago.”

The four broad categories that will be in the Republican plan, he said, are liability reforms, proposals help to ensure students can return to schools and universities in the fall, jobs and health care.

Senate Labor-HHS-Education Appropriations Subcommittee Chairman Roy Blunt, R-Mo., said last week that staff were drafting legislative text that would help schools get the large volume of COVID-19 tests needed to bring students back to classrooms and campuses later this year.

“The most important thing we do to resume normalcy is to get people back to school. You’re not going to do that, particularly in a residential setting, without millions of tests that people can take dozens of times,” Blunt said. “That’s very practical, it’s very possible.”

The House passed a $3.5 trillion COVID-19 relief package in mid-May on a mostly party-line vote, and Democrats have since been urging Republicans to begin bipartisan talks.

That bill would provide nearly $1 trillion in additional aid to state and local governments as well as billions in funding for education, health care, rental assistance for low-income households, mortgage payment assistance, a boost for the Postal Service and more.

House Democrats included funding for another round of direct payments in their legislation but opened up those payments to anyone with a taxpayer identification number, a move that would likely allow undocumented immigrants to receive the payments. The bill also increased the age for direct payments for children up to 24 for full-time students.

McConnell at the time called the Democrats’ 1,815-page bill a “liberal wish list.”

Niels Lesniewski contributed to this report.