WASHINGTON—Senate negotiators agreed on a $4.6 billion spending bill to address the rise in the number of people entering the U.S. at the southern border, advancing a compromise package more than a month after the Trump administration requested the money.

House Democrats are still reviewing it and haven’t signed onto the deal, which Senate Appropriations Chairman Richard Shelby (R., Ala.) and Sen. Patrick Leahy (D., Vt.), the top Democratic appropriator in the Senate, reached on Tuesday. The bill cleared the Senate Appropriations...