As the House returns from recess, a potential impeachment inquiry looms over Congress, with Speaker McCarthy calling it “a natural step forward” in the House’s investigation of the president’s son, Hunter Biden. Yet the speaker must navigate division in his conference over impeachment, with the White House playing offense and national polls showing Americans are skeptical.

For Republicans on the right flank of their party, impeachment has become a major priority, even as the House must approve major funding bills or face a government shutdown. House conservatives like Congresswoman Marjorie Taylor Greene are demanding that the House impeach President Biden as a condition of funding the government.

“I will not fund the government unless we have passed an impeachment inquiry on Joe Biden,” Ms. Greene said at a Floyd County town hall in Georgia. “There should be no funding for Jack Smith’s special counsel.”

Ms. Greene also said that the House should withhold funding for law enforcement working on the various investigations and prosecutions of President Trump, and she wants the House to withhold funding for Ukraine’s defense.

Mr. McCarthy, in anticipation of having issues passing all 12 funding bills needed by December, has already warned the Republicans most eager to impeach Mr. Biden that a government shutdown would complicate the process.

“If we shut down, all of government shuts it down — investigations and everything else — it hurts the American public,” Mr. McCarthy told Fox News last week.

At the same time, it’s not clear that Mr. McCarthy has the 218 votes he would need to ensure that any vote on an impeachment inquiry gets passed, as some moderate Republicans appear resistant to the idea of impeaching Mr. Biden without stronger evidence of misdeeds.

Publicly, some members of Congress have said they don’t think the investigations have turned up the sort of concrete evidence of wrongdoing that would merit impeachment.

Congressman Don Bacon told NBC in August that he thinks “it’s too early for a formal impeachment inquiry.” He added that “we should have more confidence that actual high crimes and misdemeanors occurred before starting a formal impeachment inquiry.”

The Oversight Committee chairman, James Comer, has insisted that Mr. Biden “participated in his family’s influence peddling schemes.”

The dynamic is one that has become familiar in this Congress. Mr. McCarthy is walking a tightrope, needing to accommodate the most conservative members of his conference while also protecting the most moderate Republicans who are vulnerable in 2024.

While there’s no set timeline for an impeachment inquiry vote, it would likely come before September 30, if it is going to be attached to government funding bills. The House will need to pass appropriations bills by September 30 in order to avoid a shutdown.

Democrats, including Mr. Biden, have already seized on the opportunity to portray Republicans as being disconnected from their own constituents, focusing on investigating the president and his family instead of addressing Americans’ material concerns.

In a memo obtained by Politico, which is supposed to be sent out tomorrow, Deputy Press Secretary Andrew Bates attacks House Republicans for their rejection of funding levels agreed to in the debt ceiling deal, saying “they need to remember that lives are at stake in the government funding debate.”

In the memo, Mr. Bates highlights Mr. Biden’s call for Congress to provide $800 million in funding to combat fentanyl trafficking as well as statements calling for a crackdown on trafficking from prominent Republicans like Mr. McCarthy and Congresswoman Elise Stefanik.

“House Republicans have a pressing choice to make: do they keep their commitment to the country and provide the resources we need to address our urgent challenges, or do they set back the most critical needs of the American people by deciding that a baseless impeachment stunt demanded by Marjorie Taylor Greene and their most extreme members is somehow more important?” Mr. Bates asked.

The memo is emblematic of the sort of messaging that will likely keep coming from the White House and its Democratic allies as Republicans head toward an impeachment that the White House has said will backfire on Republicans.

While House Republicans and the White House battle to set the narrative around the looming impeachment inquiry, the majority of the American public appears to be more sympathetic to the White House despite Mr. Biden’s low approval rating.

In a recent Wall Street Journal, GBAO, and Fabrizio Lee and Associates poll, 41 percent of respondents were in favor of impeachment, with 52 percent opposed. Another 7 percent didn’t have an opinion or refused to answer the question.

Broken down further, only 32 percent of respondents were strongly in favor of impeaching Mr. Biden, and 9 percent were “somewhat” in favor. For comparison, 46 percent were strongly opposed to impeaching Mr. Biden, while 7 percent were “somewhat” opposed.