After weeks of flagging negotiations, the White House has floated a proposal to allow border officials to expel migrants without asylum screenings on days when border crossings are particularly high, senators and aides said.

Key details remain to be fleshed out, including the number of migrant crossings that would trigger that new power. Sen. Thom Tillis (R-N.C.), who has been involved in the negotiations, said Wednesday that he believed the authority should kick in on days when crossings reach 3,000 or less, while some Democrats prefer a higher number of 5,000 crossings or more before agents can turn away migrants.