Trump, who has promised to issue his first veto if the Senate approves the resolution, declared an emergency last month in a bid to fund a wall along the border without congressional approval.
While the Senate appears unlikely to muster the two-thirds majority needed to override a veto, a passage in the Republican-controlled chamber would be an embarrassment for the president, who has failed over more than two years in office to persuade Congress to fund his wall, a central promise of his presidential campaign.
"I can’t vote to give the president the power to spend money that hasn’t been appropriated by Congress," Paul said in a speech on Saturday at Western Kentucky University, according to the Bowling Green Daily News.
"We may want more money for border security, but Congress didn't authorise it. If we take away those checks and balances, it's a dangerous thing," he added.
Paul's office did not immediately respond to a request for comment on the report.
Trump's demand for $5.7 billion in funding for his "great, great wall" triggered the longest-ever partial US government shutdown in December and January. The government reopened but Trump later declared a national emergency in an effort to obtain funds Congress had approved for other purposes.
Rand became the fourth Republican senator to publicly back the resolution. Assuming all Democrats and the two independents who caucus with them vote in favour of the resolution, it would clear the Senate and end up on the president's desk. Republicans hold 53 of the 100 Senate seats.
An identical version of the measure has already been approved in the Democratic-controlled House of Representatives.
Democrats argue there is no border emergency and that Trump is overreaching with his declaration given that the Constitution grants Congress the power of the purse strings.
The declaration is already being challenged in the courts.
Trump has warned Senate Republicans not to vote for the resolution, saying that doing so would put them in "great jeopardy" politically with voters.