At left, Senate Minority Leader Mitch McConnell, and, at right, Kevin McCarthy, the House speaker.
CNN  — 

Senate Republicans say they’ll keep pushing for more funding for Ukraine and defense even in the wake of Kevin McCarthy’s comments that additional funding would face long odds in the GOP-controlled House, potentially setting off a clash between the House Speaker and Senate Minority Leader McConnell.

“My assumption is if when the time comes there would be a lot of collaboration, coordination and trying to make sure that whatever we do would enjoy broad support in both the Senate and in the House,” Republican Whip John Thune told CNN Tuesday when pushed on McCarthy’s comments. “Sometimes on issues like this the Senate leads. Sometimes the House leads, but it’s a heavier lift over there no question.”

In his opening remarks, McConnell drew a distinction from McCarthy’s Monday comments calling the agreed-upon defense spending “insufficient” and pledging to work to pass more funding.

A number of Republican senators offered McCarthy some political breathing room, arguing that the issue of Ukraine funding wasn’t something that had to be dealt with immediately given billions of dollars have yet to go out, but also warning that it was an issue that the Senate wasn’t going to ignore.

“It’s no surprise, having just sold an agreement, that he would want to stick by it at least initially,” GOP Sen. Roger Wicker of Mississippi told CNN. “What I think is that amount for national defense does not provide us with the security that we need over the next three to five years.”

On Monday, McCarthy threw cold water on a separate funding package for Ukraine, rejecting demands from Senate Republicans to pour far more money to help the country and go beyond defense funding levels set by the new debt limit law.

“I don’t think the first answer is to do a supplemental,” McCarthy told CNN Monday night.

McCarthy reiterated that “This is the most money we have ever spent on defense, this is most money anyone has ever spent on defense. I don’t think the first answer is to do a supplemental, let’s go through the (appropriations) process.”

The vast differences between McConnell and McCarthy’s views on defense funding are shaped by both the political reality for McCarthy – who is trying to build back support from his right flank after bringing a debt ceiling compromise to the floor – and the facts on the ground that the Biden administration still has billions left it can dole out on supplemental funding for Ukraine.

“I don’t think we’ll know until September at the earliest if more money is needed,” Sen. Chuck Grassley of Iowa told CNN. “Between now and September there is going to be the counteroffensive by the Ukrainians. If that is very successful, I think it will change the environment.”

But despite the fact it may not be an immediate issue, Republicans recognize that at some point McConnell and McCarthy could find themselves at odds over defense money.

“It’s gonna be a big issue,” West Virginia Republican Sen. Shelley Moore Capito warned.

“I am guessing we have our work cut out for us to do what I think we need to do, which is to continue to fund the effort (in Ukraine),” North Carolina Republican Sen. Thom Tillis said. “I think we’ll end up sorting it out. It may not be at the level many of us want.”

The comments come as some in the House have also said they were surprised by McCarthy’s line against a Ukraine supplemental, warning that it was still an issue that many Republicans backed.

“If Ukraine is out of bullets, I want to replenish the bullets,” Republican Rep. Dan Crenshaw of Texas told CNN. “That time will come eventually. We have an interest in not letting Russia win this war. It doesn’t take an exceptional amount of more resources to make sure that doesn’t happen.

A handful of defense hawks in the House also pushed back against McCarthy’s assertion, another sign of an intraparty battle over Ukraine aid in the coming months.

“If that’s needed, I think we should provide it,” longtime GOP appropriator Rep. Tom Cole of Oklahoma said.

He added he had not had a direct conversation with the speaker about it, but he argued those who don’t want to help Ukraine and view it as going above the topline defense number agreed to in the debt ceiling are “wrong.”

Republican Rep. Steve Womack of Arkansas said he was surprised to see the speaker’s comments, warning there will be a “battle” over this issue.

“I think there will be a battle between a lot of the defense hawks – particularly on the Senate side – and some on our side,” Womack said

Womack said that while he was surprised to hear the speaker draw a red line on a supplemental, he argued there is a political reality for McCarthy.

“If you kind of think through it, he just went through 15 rounds to be elected speaker, to get the pushback he got particularly on the rule and then on the underlying bill on the debt ceiling program, the speaker still has to be cognizant of the divisions within the conference,” Womack said.

A key McCarthy ally, GOP Rep. Dusty Johnson of South Dakota, told CNN Tuesday that McCarthy believes putting a supplemental on the floor right now would be “a violation of the deal that he’s cut with his membership and the President of the United States.

“I know the Senate thinks they are going to do something different,” Johnson said. “The reality is it is going to be very difficult for that to get enough support in the House for it to find its way on the floor.”

CNN’s Melanie Zanona, Haley Talbot and Manu Raju contributed to this report.