Sen. Rand Paul’s call to have an immigration markup in the Homeland Security panel isn’t getting much traction.

Homeland Security and Governmental Affairs ranking member Tom Coburn of Oklahoma agreed with his counterpart from Kentucky on Tuesday, saying the panel should consider the bipartisan “gang of eight” immigration overhaul bill. Coburn said he thought the committee erred in not taking procedural steps to hold a markup.

“We should have asked for a sequential referral because so much of it is going to impact the agency that’s under this committee,” the Oklahoma Republican said at a hearing Tuesday.

But Chairman Thomas R. Carper, D-Del., said in a brief interview that any sequential referral, under Senate rules, would need to get unanimous consent on the Senate floor, meaning that any single senator could object to such a request.

Carper referenced Sen. Ted Cruz’s Monday evening objection to getting to a budget resolution conference as an example. At the Tuesday hearing, Carper said he would look into the referral question with leadership.

“Needless to say, the many questions relating to this legislation, not only the ones raised by the events in Boston, are more numerous than can be suitably included in this letter,” Paul wrote in a Monday letter to Majority Leader Harry Reid, D-Nev.

“I believe the immigration bill should also be amended and refined by HSGAC,” Paul wrote. “Having HSGAC also markup the immigration reform bill will only make the bill stronger and our nation safer.”

The only expected markup of the immigration bill is set to kick off Thursday at the Senate Judiciary Committee. Senators are filing amendments in advance of that marathon event; some of the first amendments released address local concerns.

For instance, Chairman Patrick J. Leahy, D-Vt., filed an amendment Monday to bar border crossing fees, a particular issue in Vermont where people routinely drive to and from Canada. Freshman Democrat Mazie Hirono got in on the act with a series of amendments she says are important to Hawaii.

“These changes would pump millions into Hawaii’s economy by boosting foreign tourism, make good on our promises to those who served our nation and fix a longstanding competitive disparity between mainland and Hawaii fishermen,” Hirono said in a statement.