"I think we'll have a deal out of the White House, the Senate and the House by Sept. 28," the last Thursday of the month, Norquist, who is also ATR's president, told CNBC's "Squawk Box." "The White House and the House and Senate leadership have made it clear they're working together to write this bill." He also contended a breakthrough on taxes would be too big for a Friday announcement.
While working on tax reform, Republicans on Capitol Hill should also continue to try to get rid of Obamacare, Norquist added.
"We can walk and chew gum at the same time," he said. "One does not interfere with the other. These are on two separate tracks. There's a trillion dollars of tax hikes in Obamacare, a trillion dollars over each decade. The more of those tax hikes you can get rid of when you reform and replace Obamacare, the better off you are for lower tax rates."