Sen. Rand Paul plans to support a filibuster of the Senate immigration bill unless it grants Congress more border security oversight, the Kentucky Republican said on Thursday.

Paul does not plan to stand on the Senate floor for 13 hours straight like he did in March to protest President Barack Obama's drone policy, but he will withhold support for a motion to end debate on the bill, a procedural tactic that effectively could block the bill from seeing a final vote.

"Unless they change the bill, I will vote on the side of not ending the debate, which is essentially like a filibuster, but it's not the filibuster people think of," Paul said during an interview on the "Andrea Tantaros Show."

On Wednesday, the Senate rejected an amendment to the immigration bill proposed by Paul that would require the Congress to vote on whether members deem the U.S. borders "secure" every year for five years and mandate the construction of a fence along the border with Mexico. All eight members of the bipartisan group of lawmakers who wrote the original bill voted against Paul's proposal. As the bill is currently written, agencies under the executive branch will determine border security, not Congress.

Paul said he "can't imagine" that proponents of the immigration bill will succeed in persuading him to support the final bill if more border security amendments are rejected.

"If it got stronger, I could consider it, but since they rejected my call to have Congress involved with determining whether the border is secure, I can't imagine how they can get me back unless they come back to me and say, 'We've changed our mind,'" he said. "We would like Congress to be involved in this."