Republican Sen. Ron Johnson told CNBC on Thursday he wants to fix the GOP tax reform bill so he can support it.

However, he said he doesn't have enough information about the measure to even propose an amendment to change what he doesn't like.

"In the current form, I wouldn't vote for it," Johnson said on "Squawk Box," a day after becoming the first Senate Republican to explicitly say he would not back the plan.

"It's not a real good process," the Wisconsin lawmaker said. "The House is voting on theirs today. We're going to be voting on ours right after Thanksgiving."

After another day of debating the Senate tax plan, the Senate Finance Committee is expected to vote on whether to advance their measure on Friday.

Johnson said he's been working for months behind the scenes to make changes, but he added that he's not going to let his "version of perfect" sink tax reform. "I want to get this thing fixed, and vote for pro-growth tax reform that makes all American businesses competitive globally," he said. "I care deeply about this country, I care deeply about this deficit."

As a former small business owner, Johnson said he's particularly concerned about the so-called pass-through rate, in which the profits and losses of sole proprietorships, partnerships, and S-corporations "pass through" to their owners who are then taxed at individual income-tax rates, currently as high as 39.6 percent.

"We can't leave anybody behind, which is why they came up with the 25 rate for pass throughs," he said. "The problem is, neither the House or the Senate version really honored that commitment to pass-through businesses, which I argue are a huge engine of economic growth."

"I don't have the information on how much it would cost, how many pass-through businesses are being left behind that do compete globally. I can't get the information. I've been asking. They don't give it to me," said Johnson, chairman of the Senate Homeland Security Committee.

Johnson said President Donald Trump called him on Wednesday night about pass throughs and told him that he's "meeting with Treasury officials today."

"'I will work my tail off over the weekend to fix this problem,'" Johnson quoted Trump as saying.

Johnson appeared on CNBC before the Joint Committee on Taxation released on Thursday its latest look at the GOP Senate bill.

Republicans hold only a two-seat majority in the Senate, so the GOP has little margin for defections because Democrats are uniformly opposed to the plan. Sens. Susan Collins of Maine, Bob Corker of Tennessee and John McCain of Arizona have also voiced concerns over the tax plan. They refused to say whether they would ultimately vote for the bill.

The Senate's tax plan, unveiled last week, includes cutting the corporate tax rate from 35 percent to 20 percent and making broad tweaks to the individual tax system.

— Reuters contributed to this report.