Sen. Tom Coburn (R-Okla.) will insist that any federal aid to deal with the tornado in his home state must be offset by budget cuts.

(J. Scott Applewhite/AP) (J. Scott Applewhite/AP)

"He will ask his colleagues to sacrifice lower priority areas of the budget to help Oklahoma," spokesman John Hart said. Should other Republicans join Coburn, it could set up a fight similar to the January tug-of-war over Hurricane Sandy funding. That aid package was delayed by GOP opposition and ultimately passed with mostly Democratic support.

In a statement, Coburn said that "as the ranking member of Senate committee that oversees FEMA, I can assure Oklahomans that any and all available aid will be delivered without delay." He later told CNN that it was "insensitive to even talk about" budgeting for relief funding now. "It just shows the crassness of Washington versus the sensitivity that we need to have," he said.

We don't yet know what a congressional relief package for Oklahoma would look like, if one is even necessary. As of Tuesday morning, FEMA has $11.6 billion in its Disaster Relief Fund.

Coburn was against the Sandy relief package, as well as 2011 legislation to replenish the Federal Emergency Management Agency's disaster fund. His office has noted that the 1995 aid for victims of the Oklahoma City bombing was balanced by cuts to unspent appropriations. However, he did ask for expedited FEMA aid in 2007, when an ice storm hit his state.

President Obama declared a major federal disaster late Monday and ordered federal aid to deal with the Moore, Okla. tornado, which devastated an area a mile wide and 20 miles long. Coburn is also headed back to Oklahoma to asses the damage.

House Republican leaders declined to address whether they would demand spending cut offsets. "We will work with the administration to make sure that they have the resources they need," Speaker John A. Boehner (R-Ohio.) told reporters Tuesday morning.

"We will help them rebuild," Majority Leader Eric Cantor (R-Va.) said. The offsetting cuts issue first reached controversy in 2011, when an earthquake centered in Cantor's district roiled the East Coast. At the time Cantor sided with conservatives suggesting the emergency aid should be financed by other cuts to the federal budget. When it came to the Sandy aid package, Cantor took the lead in supporting the legislation.

Boehner was joined at the weekly leadership press conference by members of the Oklahoma delegation.

Sen. James Inhofe (R-Okla.) also voted against Sandy aid.

Two of Oklahoma's five members in the House of Representatives, all Republicans, voted against it. Rep. James Lankford voted for a smaller federal flood relief package but against the bigger package.

Rep. Tom Cole (R-Okla.) said on MSNBC Tuesday morning that one reason he supported the Sandy relief was Oklahoma's history of bad tornadoes.

"Frankly, one of the reasons that we try to be sympathetic to people in other parts of the country" is that "we're always one tornado away from being Joplin," he said, referencing the catastrophic 2011 tornado in Missouri. "I didn't think it was going to be quite this soon."

Ed O'Keefe contributed to this report.