Sen. Ted Cruz expects tax reform to get done "late this year or early next year" — not sticking to the hard 2017 deadline set out by top Republicans.

The Texas Republican told CNBC on Friday it will take "at least a couple months" to iron out differences within the GOP, which has a narrow majority in the Senate.

"I do think virtually every Republican wants to get to yes" on overhauling the tax system, Cruz said on "Squawk Box."

President Donald Trump, top White House officials and House Speaker Paul Ryan have all set a target of this year to approve a tax bill. Ryan said Thursday he would keep the House in session through Christmas if necessary.

Late last month, Republicans released a framework for the tax proposal they hope to pass this year. It calls for major reductions in household and corporate income tax rates and a doubling of the standard deduction.

The framework left out many key details. The congressional tax-writing committees are expected to spell out specifics once Congress passes a joint budget, a key step in the process.

Already, the GOP has run into political hurdles. Some GOP lawmakers, like Sen. Bob Corker, have expressed concerns about the potential budget deficits generated by the cuts.

Cruz has no such concerns, saying he wants an "unapologetic tax cut." Next week, the Senate is expected to pass a budget resolution allowing for $1.5 trillion in tax cuts under the reconciliation process. That number does not include the estimated effects of economic growth.

Cruz contended Republicans should go "much bigger and much bolder" on cutting taxes, saying deeper cuts will spark more economic growth and make up for the lost revenue.

Republican lawmakers in high-tax blue states have also started to push back on a proposal to get rid of state and local tax deductions, a method to raise money to help offset tax cuts.

Cruz supports ending the deductions as long as middle-income people in the states see an overall reduction under the plan.

"We can end that deduction if we're lowering the tax rate enough that even people in those blue states are seeing a net tax reduction," he said.

One independent analysis of the GOP tax framework estimated more than a quarter of middle-class Americans would eventually see a tax increase under the changes. Sen. Rand Paul, R-Ky., Cruz's fellow fiscal conservative, raised concerns about tax cuts not being deep enough after the release of that estimate.

Cruz said the question he wants to ask as the GOP drafts the plan is, "Are we improving the lives, are we improving the take-home pay, are we improving the prospects of working men and women?"

Cruz waged a fierce battle against Trump for the 2016 GOP presidential nomination.