By Lindsay Wise and Siobhan Hughes

WASHINGTON -- Republican senators splintered into factions over their party's proposal for the next coronavirus-aid package, as GOP lawmakers' antipathy toward government spending ran into the turbulence of election-year politics.

With the country in the grip of a pandemic, the party is divided over the scope and scale of aid needed to keep the economy afloat -- and over whether their party's precarious standing in polls justifies setting aside ideological concerns about the fast-growing federal deficit fewer than 100 days before voters head to the polls.

"You can see a bunch of people already pushing back," Sen. Lindsey Graham (R., S.C.) said. He added that if Senate Majority Leader Mitch McConnell (R., Ky.) can get half of Senate Republicans to vote for the bill, "that'd be quite an accomplishment."

At a press conference on Tuesday, Mr. McConnell acknowledged the "fulsome discussion" in the party.

"I think it's a statement of the obvious that I have members who are all over the lot on this," Mr. McConnell said, who called the plan a starting point that represented the broadest consensus of GOP senators. "There are some members who think we've already done enough, other members who think we need to do more."

The stimulus debate pits the GOP's political pragmatists against its spending hawks, with the fate of swing-state incumbents hanging in the balance: At-risk Republican senators don't want to return to the campaign trail during the August recess empty-handed, while fiscal conservatives recoil at any plan that they see as ballooning the deficit and conditioning the public to expect broader government assistance once the pandemic is over.

At stake could be control of the Senate and White House, some Republicans warn. The nonpartisan Cook Political Report last week released a new analysis of key Senate races that for the first time this cycle favored Democrats to take back the chamber. Democrats are expected to keep control of the House this fall.

The situation complicates the party's negotiations with Democrats, which Mr. McConnell said would be led by White House chief of staff Mark Meadows and Treasury Secretary Steven Mnuchin.

Some Republicans are grumbling privately that Messrs. McConnell and Mnuchin waited too long to start putting a bill together and the process hasn't been transparent. They are worried that the GOP's opening $1 trillion bid puts them at a disadvantage in talks with Democrats, who have proposed a rival $3.5 trillion plan.

Those complaints boiled over at a closed-door lunch on Tuesday, where arguments broke out between GOP senators who sparred over major elements of the Senate GOP plan unveiled on Monday by Mr. McConnell. The Heals Act would fund another round of direct payments to Americans, assistance for schools and small businesses and provide liability protections against coronavirus lawsuits.

The proposal also provides more money for testing, contact tracing and vaccines and extends unemployment benefits, although it would cut the current federal $600 weekly supplement to $200 a week through September.

Senate Minority Leader Chuck Schumer (D., N.Y.) called Republicans and their proposal "ununified, unserious and completely unsatisfactory," in remarks on the Senate floor Tuesday. Following a meeting with GOP negotiators, Democrats also criticized the liability shield, which Mr. McConnell has called a must-have in the bill.

"What the leader said sounded like someone who has no interest in having an agreement," said House Speaker Nancy Pelosi (D., Calif.).

The plan received unenthusiastic reviews from some Republicans as well.

"I think it would be fair to say at this juncture that we have unity in disagreement," quipped Sen. John Kennedy (R., La.). "It's a mess," said Sen. Josh Hawley (R., Mo.), who added that he wasn't inclined to vote for it.

"It's a trillion dollars," complained Sen. Rick Scott (R., Fla.), emphasizing the word trillion. "That's 5% of our GDP!"

One of the most vocal critics of the proposal was Sen. Rand Paul (R., Ky.), who stormed out of the GOP lunch exclaiming that he must have been at the wrong meeting. "Trillions of dollars of debt! This is what Republicans stand for?" he said.

Mr. Paul said some of his colleagues who are running for re-election -- and Republican leaders trying to protect them -- are misguided when they say that the party must spend now or lose in November.

"I tell them: What if you anger the base? The people who got you here, the people that are the hard-core conservatives?" Mr. Paul said. "They can go vote for the libertarian, they can vote for independents, they can sit home."

Making the opposite case was Sen. Tom Cotton (R., Ark.), who told his Republican colleagues at last week's lunch that they should pay attention to what the vulnerable Senate Republicans needed from the package, even if it meant more spending. He argued that if Democrats won the Senate and Democratic candidate Joe Biden won the presidency, they would pass much more expensive legislation, and that the GOP should take steps now to avoid such an outcome.

President Trump and Mr. Mnuchin later called Mr. Cotton to thank him for making the argument, according to a Republican familiar with the conversation. The White House didn't respond to a request for comment.

GOP Sen. Thom Tillis, locked in a tough re-election fight in the battleground state of North Carolina, said Republicans need to pass something fairly soon, especially to extend unemployment benefits, which are set to expire on July 31.

"We're going to have to recognize this virus is going to stick around a lot longer than anybody expected," Mr. Tillis said.

But conservative advocacy groups and activists have mobilized to push back hard against the plan from Messrs. McConnell and Mnuchin. The Club for Growth, a conservative advocacy group, started telling lawmakers privately Wednesday that it planned to urge a "no" vote against Mr. McConnell's bill, and would penalize any senators who voted for it in the group's high-profile congressional scorecards. That was a change from the group's relative silence on previous coronavirus-aid packages.

"I think the president has been badly served by Mnuchin and McConnell when they just decided we're going to run another one of these big spending bills," said David McIntosh, the Club's president. "And we're saying very publicly now that if McConnell has his way, that Republicans will lose, that he will have led the Republicans into the minority in the Senate...Voters will say, 'Why elect Republicans when we can get Democrats to give us handouts?'"

Asked whether Republicans' disagreements will weaken their party's hand with Democrats, Sen. John Cornyn (R., Texas) countered that conservative opposition to additional spending could help keep down the price tag of the eventual deal.

--Andrew Duehren contributed to this article.

Write to Lindsay Wise at lindsay.wise@wsj.com and Siobhan Hughes at siobhan.hughes@wsj.com