"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."
Last week, the so-called skinny Obamacare repeal failed in the Senate, with three GOP defections — Sens. Susan Collins of Maine, Lisa Murkowski of Alaska and John McCain of Arizona — sinking the measure in a 49-51 vote.
"All you need to get is one Senate vote, and McCain's argument for not voting for health-care reform … I don't think that stands. He needs to decide what he does want there," Norquist said, referring to McCain's call for a reset on health care and a return to an open, bipartisan process. "I think it's entirely likely they come back and get that extra vote and move forward there on some health-care reform," added Norquist.
President Donald Trump prodded Republican senators on Twitter over the weekend to use the "nuclear option," a last-ditch way for the majority party in the Senate to overcome obstruction by the minority party by forcing a simple majority vote. Republicans hold a slim a four-seat advantage over the Democrats, with Vice President Mike Pence waiting in the wings to break any ties.
In May, after fits and starts, the House narrowly passed its version of legislation to repeal and replace Obamacare in a 217 to 213 vote. That was an even slimmer margin considering the GOP holds a much larger 47-seat majority in the House.