WASHINGTON—Senate Republicans are warning President Trump against moving forward with his national-emergency declaration to build a southern border wall and pressuring him to restrict the project’s funding to less controversial sources.

Republicans are on the brink of attaining enough support for a resolution to disapprove of the declaration, potentially forcing Mr. Trump to use his veto power for the first time. On Thursday, Sen. Lamar Alexander (R., Tenn.) decried the president’s move on the Senate floor, seeming to warn...