Apple and other multinational companies can expect to start paying a lot more taxes in the coming years, University of Southern California law professor Edward Kleinbard said Monday.

The only question is how much of that tax revenue the United States will get and how much Europe will get, he told CNBC.

"The easy days of single-digit tax rates are going to be over," Kleinbard said in an interview with CNBC's "Closing Bell."

Apple and other multinational firms have been accused of dodging U.S. corporate taxes by going abroad. Apple has relocated much of its earnings to Ireland, allowing the company to pay a much lower corporate tax rate in some years — as low as 2 percent, compared to 35 percent in the U.S.

On Tuesday, the European Commission will rule against Ireland's tax dealings with Apple, according to Reuters.

Ireland is accused of letting Apple shelter profits worth tens of billions of dollars from tax collectors in return for maintaining jobs. Apple and Ireland rejected the accusation and have both said they will appeal any adverse ruling.

Apple CEO Tim Cook has struck back at critics of the company's tax policy, telling the Washington Post Apple would not bring its money back from abroad unless there was a fair rate.

Kleinbard agreed the tax system in the U.S. is broken.

However, "because it's broken, firms invert. Because it's broken, Apple pays extraordinarily low tax rates on its global income. The end result is going to be more tax for Apple regardless of tomorrow's ruling," he said.

And while Ireland is going to be caught in an "embarrassing" position with Tuesday's ruling, it won't be a financially difficult one, Kleinbard said.

"To the contrary, when the EC rules, it rules against Ireland technically and then it compels Ireland to capture the money back from Apple. So Ireland is going to be rolling in dough," he said.

— Reuters and CNBC's Javier David contributed to this report.