House Majority Leader Eric Cantor said House Republicans could accept a 'big deal' debt-limit increase — as long as it does not include tax hikes.

"Currently, there is not a single debt limit proposal that can pass the House of Representatives, but I believe the path forward is to focus on what we can agree upon, and though it doesn’t go as far as our budget, House Republicans can likely agree with the general spending cuts and entitlement changes in the ‘big deal’ proposed by the President," Cantor (R-Va.) said in a statement on Wednesday.

He added that a $4 trillion deal that President Obama has been pushing and House Speaker John Boehner (R-Ohio) once showed an openness to was not ideal for House Republicans.

"That isn’t everything that we want, as the President’s proposal appears to be around $3 trillion in spending cuts, compared to the reforms and the more than $6 trillion in cuts House Republicans supported in the Ryan budget," Cantor continued.

In the same statement, Cantor criticized Obama for refusing to repeal his administration's healthcare reform law, and warned that any deal that included tax increases would not pass the House.

"The President refuses to compromise on the repeal of ObamaCare, and House Republicans refuse to raise taxes, so both have been ruled out. Further, the simple reality is that tax increases cannot pass the House, and the constant demand for them makes coalescing around any increase in the debt limit less likely," Cantor said.

Cantor and the conservative wing of the Republican Party have been unwavering in their promise to not agree to a debt-ceiling increase package if it included tax increases of any kind. That's been a sticking point in the debt-ceiling negotiations, as Democrats have called for added revenues in any debt-ceiling increase deal.

Obama has shown an openness to making cuts to Medicare and Social Security programs if Republicans agreed to pass a large $4 trillion debt-ceiling increase deal that included added revenues, likely in the shape of closing tax loopholes and ending certain subsidies.