Democratic and Republican sources working to build support for the bill agreed that garnering the necessary GOP votes would require amending the legislation to strengthen its border security components. Additionally, there are senators on both sides of the aisle who don’t support the current measure because of provisions other than border security who also must be appeased.

The bill’s supporters are now focusing their efforts on wooing as many as five Democrats who might oppose it and about two dozen Republicans described as “gettable” or “maybes” but who for now are far from a “yes” vote. (Senate Minority Leader Mitch McConnell, R-Ky., is not among those targeted.) The bill is scheduled to hit the floor next Monday, and Senate Democratic leaders said it may go up for a final vote before the July 4 recess.

“They probably are close to 60, but they’re not yet at 60 right now,” a former Republican Senate leadership aide said. “My sense is, for that to happen, there probably will need to be changes to border security.”