WASHINGTON—Most Republican senators have signed on to a pledge to force Democrats to raise the debt ceiling through procedures that don’t rely on GOP votes, escalating the political tug of war over who is responsible for keeping the U.S. from defaulting.

Sen. Ron Johnson (R., Wis.) said in an interview that some 46 Republicans have signed on to a letter that he circulated during a rapid-fire series of votes on a budget resolution that kicks off Democrats’ efforts to pass a $3.5 trillion cornerstone of President Biden’s agenda.

“They shouldn’t be expecting Republicans to raise the debt ceiling to accommodate their deficit spending,” Mr. Johnson said.

The Democrats’ $3.5 trillion budget outline doesn’t include a measure to increase the U.S. government’s borrowing limit. Democrats plan to pass their antipoverty and climate plan through a process called budget reconciliation, which requires just a simple majority. In the 50-50 Senate, Democrats don’t have the 60 votes needed to pass a debt-ceiling increase through the ordinary legislative process.

Senate Minority Leader Mitch McConnell (R., Ky.) had previously said Democrats shouldn’t expect any Republican help on the debt limit, and reiterated that position Tuesday.

Budget reconciliation may offer Democrats a way to sidestep some partisan gridlock and advance President Biden’s policy objectives. WSJ explains how the process works and why divisions in the party could scuttle the process. Photo Illustration: Carlos Waters/WSJ The Wall Street Journal Interactive Edition

“This debt ceiling is going to cover all of the things that all of us have been opposing,” he said in an interview, adding the Democrats “need to do the responsible thing and raise the debt ceiling, because America must never default on its debt.”

Democrats note recent deficits have also been driven by the 2017 Republican tax cuts, as well as bipartisan relief packages during the Trump administration.

Raising the debt limit doesn’t authorize new spending, but rather allows the Treasury Department to issue new debt to cover spending that Congress has already authorized, including payments to bondholders, Social Security recipients, veterans and others.

Under current law, the Congressional Budget Office projects cumulative budget deficits will total $12 trillion over the next decade, due in large part to rising costs of automatic spending programs such as Medicare and Social Security, as well as higher interest payments on the debt. Even if lawmakers don’t enact new deficit-financed legislation, the government will need to borrow to cover those obligations Congress has already authorized.

On Aug. 1, the debt limit, which had been suspended under a 2019 law, was reinstated at around $28.5 trillion, a figure that includes debt held by the public and debt held by government agencies.

The Treasury uses emergency accounting maneuvers to conserve cash so the government can keep paying its obligations, but those measures are expected to run out some time in the fall. Unless Congress steps in to suspend the debt limit again, the Treasury could begin to miss payments on its obligations and default.

One idea Democrats are considering is a stand-alone vote to suspend the debt limit, a person familiar with the matter said earlier this week, a move that could put pressure on Republicans to support the measure or risk rattling financial markets if the vote fails. Lawmakers could also attach the measure to another must-pass bill, like government funding, to force the issue.

In the letter, the Republicans said that Democrats need to take responsibility for the consequences of their spending, including the $1.9 trillion coronavirus-relief package that passed the Senate earlier this year without any Republican support. CBO said last month that faster economic growth spurred by the relief bill will likely offset the measure’s entire cost, and estimated that deficits over the next decade will be slightly lower than last projected in February, before the bill passed.

“Democrats, at any time, have the power through reconciliation to unilaterally raise the debt ceiling, and they should not be allowed to pretend otherwise,” the letter said, adding that Republicans wouldn’t vote to boost the debt limit through a stand-alone bill or any other vehicle.

Mr. Johnson said that the idea of the letter started at a Senate Republican lunch on Monday, at which one member suggested a letter on the topic. Mr. Johnson wrote one and at the Tuesday lunch, said that he would circulate the letter during the so-called vote-a-rama. One member asked Mr. McConnell if he would sign the letter, and Mr. McConnell said that he would. His signature is the first one to appear on the letter.

The only Republicans who didn’t sign on are Sens. Lisa Murkowski (R., Alaska), Susan Collins (R., Maine), Richard Shelby (R., Ala.) and John Kennedy (R., La.)

Write to Siobhan Hughes at siobhan.hughes@wsj.com and Kate Davidson at kate.davidson@wsj.com