Norquist's group, Americans for Tax Reform, asks all Republican candidates to sign a no-new-taxes pledge, but Trump has not signed one yet. Norquist said he was confident the candidate would sign given his public comments.
"I'm very happy, I sleep very well at night," Norquist said on CNN, referring to Trump's tax stances.
Trump on Monday also sought to clarify comments he made last week about the U.S. debt. He said he never espoused restructuring or defaulting on government debt but would buy back the debt at a discount if interest rates go up.
Trump effectively clinched the Republican nomination for the Nov. 8 presidential election with a primary election victory in Indiana last Tuesday that forced his last two rivals out of the race.
His candidacy has caused a rift in the Republican Party, where many leaders have been appalled by his rhetoric on immigrants and Muslims and concerned that some policy positions such as Trump's opposition to free trade run counter to Republican orthodoxy.
Many Republicans oppose raising the federal minimum wage, but Trump has embraced a higher minimum. He said he would leave the issue up to the states, because some places, such as New York, are more expensive to live in.