The concerns expressed by Republican senators are hardly monolithic, and Mr. McConnell will have to walk a delicate line to resolve the issues without setting off additional objections from other lawmakers.
The deficit issue is a crucial one, given several senators have already expressed concerns about piling up more debt as a result of the tax overhaul. Senator Bob Corker of Tennessee has said he will not vote for a tax plan that he determines will add to the deficit, after accounting for economic growth spurred by the legislation. Senator James Lankford of Oklahoma has also expressed deficit concerns.
Mr. Flake, who has spoken positively in recent days about the effort to overhaul the tax code, repeated his deficit concerns on Monday.
“We’ve got to realize we have a $20 trillion debt and a deficit that’s about $600 billion a year, and we can’t do things that are simply going to explode that deficit,” Mr. Flake said on KFYI, an Arizona radio station.
“I think we desperately need tax reform,” Mr. Flake said. “It’s been more than 30 years since we’ve had significant tax reform. My concern is that it’s really tax reform and not just tax cuts.”
Another matter under scrutiny is how the Senate bill treats small businesses and other so-called pass-through entities, whose owners pay taxes on profits through the individual tax code.
Senator Ron Johnson, Republican of Wisconsin, objects to how such businesses would fare under the legislation, which in his view favors larger corporations. Mr. Johnson came out against the bill last week.
In an interview on the radio station WISN on Monday, Mr. Johnson expressed optimism that the issue would be addressed. But he still complained about what he deemed an “awful, rushed process” with a “desperation to pass anything.”
“That’s not the best way to pass something that’s going to affect so many people’s lives and have such a dramatic impact on our economy,” Mr. Johnson said.
Then there is the contentious issue of health care, which Senate Republicans brought into the already difficult tax debate when they added to their bill the repeal of the Affordable Care Act’s requirement that most people have health coverage or pay a penalty. The Senate proposal would eliminate those penalties, effectively scrapping the coverage requirement, known as the individual mandate.
The inclusion of the repeal immediately raised questions about how such a provision might influence the votes of the Republican senators who objected to previous efforts this year to repeal the Affordable Care Act, particularly Susan Collins of Maine and Lisa Murkowski of Alaska.
This time, Ms. Murkowski has a big reason to support the tax overhaul: The legislation is to be combined with a measure to open the Arctic National Wildlife Refuge in Alaska to oil and gas drilling, a goal she has championed.
Ms. Collins has concerns about the tax bill on multiple fronts, including on the issue of health care. She warned against adding the repeal of the individual mandate to the bill, and now that it has been included, she said Congress should also pass separate legislation to help stabilize insurance markets. The Congressional Budget Office has estimated that repealing the individual mandate would increase average health insurance premiums on the individual market by about 10 percent.
“If you do pull this piece of the Affordable Care Act out, for some middle-income families, the increased premium is going to cancel out the tax cut that they would get,” Ms. Collins said on CNN’s “State of the Union” on Sunday.
Beyond the issue of health care, for instance, she wants to keep the current top tax rate, 39.6 percent, for people making $1 million or more, rather than lower it to 38.5 percent as the Senate bill would do.
But now that the repeal of the individual mandate has been added to the bill, taking it out to pacify someone like Ms. Collins would create its own problems. Getting rid of the mandate is projected to save the government more than $300 billion over a decade, and Senate Republicans are counting on those savings to offset some of the cost of their tax cuts.
Still, the president’s budget director, Mick Mulvaney, said on Sunday that the administration would be open to dropping the repeal of the individual mandate from the tax overhaul if necessary.
“If we can repeal part of Obamacare as part of a tax bill, and have a tax bill that is still a good tax bill that can pass, that’s great,” he said on CNN. “If it becomes an impediment to getting the best tax bill we can, then we’re O.K. with taking it out.”
At the White House on Monday, Mr. Trump expressed optimism about the tax effort, congratulating the House for passing its bill last week and expressing hope that the Senate would soon follow suit.
“We’re going to give the American people a huge tax cut for Christmas,” the president said. “Hopefully that will be a great, big, beautiful Christmas present.”