Paul is pressing the administration for greater clarity on whether the Obama administration has the authority to use lethal force, such as a drone, against a suspected terrorist who is a U.S. citizen.
"Do you have the authority to kill Americans on American soil?" Paul summed up the question for reporters on Thursday. He said he had not received a response from the White House.
Hours after the filibuster, Republican leader McConnell said Paul deserves an answer.
"It simply doesn't have that right, and the administration should just answer the question," McConnell said. "There is no reason we cannot get this question answered today, and we should get this question answered today. Frankly, it should have been answered a long time ago."
At the White House, spokesman Jay Carney said the attorney general sent Paul a letter Thursday afternoon answering the senator's question about whether drones could be used against U.S. citizens on American soil. Carney, quoting from the letter, said: "Does the president have the authority to use a weaponized drone to kill an American not engaged in combat on American soil? The answer is no."
Carney said White House officials have also been in touch with Paul's office.
The Obama administration has said it has not conducted such operations inside U.S. borders, nor does it intend to. Paul and backers said that wasn't good enough. They wanted the White House to rule out the possibility of them happening altogether.
Senate Majority Leader Harry Reid, D-Nev., said the Senate would vote to end the filibuster Saturday morning unless lawmakers can reach an agreement to vote earlier.
Paul's performance, which centered on questions about the possible use of drones against targets in the United States, clearly energized a number of his GOP colleagues, who came to the floor in a show of support and to share in the speaking duties. And even as the night progressed, Paul appeared invigorated despite being on his feet for so long. Actual talking filibusters have become rare in the Senate, where the rules are typically used in procedural ways to block the other party's agenda.