WASHINGTON • After meeting at the White House, President Joe Biden and a group of Republican senators agreed to talk again early next week as negotiations intensified Thursday over a potentially bipartisan infrastructure package that could become one piece of the administration’s ambitious $4 trillion public investment plan.

The GOP senators exited the more than 90-minute meeting “encouraged” about their discussions with the president and prepared to build on the $568 billion proposal they had put forward last month as an alternative to his sweeping American jobs and families plans.

“The president asked us to come back and rework an offer so that he could then react to that,” said Sen. Shelley Moore Capito of West Virginia, who is leading the group.

“We’re very encouraged,” she told reporters outside the White House. “The attitude the president had in the Oval Office with us was very supportive and desirous of striking a deal.”

Biden also emerged upbeat. “I am very optimistic that we can reach a reasonable agreement — and even if we don’t it’s been a good faith effort,” Biden said in the Rose Garden.

Biden is intent on at least trying to strike a deal with Republicans rather than simply going it alone with a Democrats-only bill, which might in some ways be a more politically viable route in a Congress held by the president’s party with only the slimmest of majorities.

One strategy that appears to be coming into focus would be for Biden to negotiate a more limited, traditional infrastructure bill of roads, highways, bridges and broadband as a bipartisan effort. Then, Democrats could try to muscle through the remainder of Biden’s priorities on climate investments and the so-called human infrastructure of child care, education and hospitals on their own.

“I’m willing to negotiate,” Biden said earlier at the White House. But the president has indicated that he’s not about to wait indefinitely for a compromise that may or may not come, and reiterated his view Thursday that “doing nothing is not an option.”

The White House said the president stressed that inaction was a “red line for him.” He set a Memorial Day deadline for progress on a bipartisan deal.

Those gathered included some of the top ranking Republicans — Sens. John Barrasso of Wyoming, Roy Blunt of Missouri, Mike Crapo of Idaho, Pat Toomey of Pennsylvania and Roger Wicker of Mississippi. Joining Biden were Vice President Kamala Harris, Transportation Secretary Pete Buttigieg and Commerce Secretary Gina Raimondo.