Speaker McCarthy’s unilateral launch of an impeachment inquiry into President Biden on Tuesday — a move long sought by the right flank of the House GOP conference — has yet to solve a single one of his political problems, as demands for steep spending cuts, including defunding the Department of Justice and ending aid to Ukraine, ramp up just days before a potential government shutdown. 

Speaking at a press conference outside of his office at the Capitol on Tuesday, Mr. McCarthy said, “House Republicans have uncovered serious and credible allegations into President Biden’s conduct. Taken together, these allegations paint a picture of a culture of corruption.”

“These are allegations of abuse of power, obstruction, and corruption, and they warrant further investigation by the House of Representatives,” he added. “That’s why today, I am directing our House committees to open a formal impeachment inquiry into President Joe Biden.”

Republicans in Congress, especially those most infatuated with President Trump, have been demanding an impeachment vote for Mr. Biden for more than two years. Congresswoman Marjorie Taylor Greene first filed articles of impeachment on January 21, 2021 — one day after Mr. Biden was sworn in.

On Wednesday, the New York Times reported that Mr. Trump dined with Ms. Greene on September 9 to discuss the push for impeachment. The Times also reported that Mr. Trump has kept in regular contact with congressional allies, urging them to support the measure. 

Mr. McCarthy’s move to open an impeachment inquiry on the same day the House returned from its six-week summer recess comes as he attempts to negotiate a budget that would satisfy his most conservative colleagues, the Democratic-controlled Senate, and Mr. Biden. 

The inquiry, though, has not won him any favors with his most ardent detractors. At a press conference on the grounds of the Capitol just hours after the inquiry was announced, members of the conservative House Freedom Caucus gathered for a press conference to make it clear they would not lift a finger to help Mr. McCarthy. 

With the October 1 deadline for a government funding deal looming, members of the Freedom Caucus told reporters that they would not support any short-term extension — known as a continuing resolution — in order to keep the government open so that Congress can work on next year’s budget. 

“Under no circumstances should we have a dirty CR or a clean CR, so-called, on October 1 — that should not be on the table,” a rising star in Republican politics who challenged Mr. McCarthy for the speakership in January, Congressman Chip Roy, said.

“We’re not interested in a continuing resolution that continues the policies and the spending of the Biden-Schumer-Pelosi era and we’re not going to vote for it,” the chairman of the Freedom Caucus, Congressman Scott Perry, said.

Joining the Freedom Caucus in its demand for big spending cuts in any hypothetical continuing resolution is the Republican Study Committee, the largest caucus in the House GOP, with 156 of the House’s 222 GOP members.

Ukraine aid has taken center stage in the government funding negotiations, with Senator McConnell making clear that a sizable chunk of his GOP colleagues in the upper chamber will agree to Mr. Biden’s request for more than $20 billion in additional aid to the besieged country. Fox News first reported on Tuesday that the director of the Office of Management and Budget, Shalonda Young, had sent a letter to congressional Republicans estimating that the federal government has spent more than $100 billion on aid to Ukraine. 

“Since Putin’s escalation in Ukraine, President Biden has not been as decisive as many of us would have preferred,” Mr. McConnell said on the Senate floor on September 6. “But this is no excuse for Congress to compound his administration’s failures with failures of our own,” he added, urging his colleagues to send billions more to the besieged nation. 

A significant number of House conservatives will likely avoid voting for any bill that includes such aid. Earlier this summer, Congressman Matt Gaetz proposed eliminating all aid to Ukraine. The legislation ended up getting 70 votes on the House floor. 

In a sign of how likely a government shutdown is on October 1 and the desperation of his negotiating position, Mr. McCarthy proposed on Wednesday a spending deal that would keep the government open temporarily while fully funding military construction and veterans affairs programs, the Department of Homeland Security, and disaster aid for a string of recent devastating weather events. 

A former chairman of the House Budget Committee, Congressman Steve Womack, told the Hill that Mr. McCarthy’s plan is “trying to get the conference to coalesce around something that we can all come to agreement on.” He added that he and his colleagues “ought to be able to do this post haste and move in a positive direction. It also sends a signal to the Senate that we’re serious about what we’re doing and we have some solutions.”

On the other side of the Capitol, Republicans and Democrats in the Senate have already begun work on the 12 appropriations bills that Mr. Biden will have to sign to fund the government fully. The members of the upper chamber are doing it in a bipartisan manner. 

The Senate is planning to vote on three of the 12 funding bills by the end of the week. The chairwoman of the Senate Appropriations Committee, Senator Murray, made clear that, unlike members of the House, she is working closely with her Republican colleagues to craft bipartisan legislation. 

“It is important to note, as we wrote these bills during the committee process, I made certain to work in lockstep with my counterpart, Vice Chair Collins,” Ms. Murray said, referring to her Republican colleague of Maine. “Together, we collaborated closely with members on both sides of the aisle on and off the committee to get input from nearly every senator.”