In his first major interview since being elected the first Black leader in Congress, the New York Democrat Hakeem Jeffries was asked what he thought of Kevin McCarthy, the Californian now seeking the votes to be speaker when Republicans take over the House next year.

“We serve in Congress together,” Jeffries said.

Then, CNN reported, the new House minority leader “stopped talking”.

Given the rancorous nature of US politics, particularly in the House of Representatives, Jeffries’ reluctance to speak warmly of his opponent, or even to comment at all, was not particularly surprising.

Pressed, he said: “I respect the fact that [McCarthy] is the current House Republican leader, and depending on what happens on 3 January, may be the next Republican speaker.”

McCarthy’s party took the House in last month’s midterm elections but not with the “red wave” many expected, the result a narrow majority and a would-be speaker at the mercy of the pro-Trump far-right.

Jeffries said: “It’s incredible to me that even at this point in time, as [Republicans are] on their way temporarily into the majority, they have not articulated a vision for addressing the economic concerns of the American people. It’s because there’s a real risk that the incoming Republican majority is being hijacked by the extremists who have grown in ranks.”

On Sunday, CNN asked Mike Lawler, a Republican congressman-elect from New York, a state where the party performed relatively well, if he would back McCarthy.

Lawler said: “We’re not going to be held hostage by a handful of members when the overwhelming majority of the conference is in full support of Kevin McCarthy.”

Speakers need only 218 votes, a simple majority, regardless of party lines. The longest such contest, concluding in February 1856, went through 133 ballots. Lawler said he would back McCarthy through numerous rounds if necessary.

“I will only be voting for Kevin McCarthy for speaker,” he said. “I know many of my colleagues within the conference feel the same way. This is potentially something that could come to a head, but I do think cooler heads will prevail and I do think on 3 January, Kevin will have the necessary votes to become speaker.”

Also on Sunday, Jeffries told ABC’s This Week his mission would be “to find ways to work with Republicans whenever possible to get things done for the American people … but we will also oppose them when we must, particularly as it relates to any effort to go down this rabbit hole of unnecessary, unconscionable, unacceptable investigations of the administration.”

House Republicans have indicated targets for investigation will include Joe Biden and his son Hunter Biden, immigration policy and the House committee investigating Trump’s election subversion and the January 6 Capitol riot.

Jeffries said he had “not had any conversations with Republicans yet. We are in the process of organising as Democrats. They are in the process, of course, of organising as Republicans. But I look forward to those conversations, certainly.”

He was also asked about suggestions that a moderate Republican challenger to McCarthy might attract enough Democratic votes to become speaker. Jeffries hedged, saying, “I think the question right now is, what are the Republicans going to do?” Pressed on the matter, asked if the door was “still open” to such a scheme, he said: “Well, let’s see.”

Dave Joyce of Ohio, chair of the moderate Republican Governance Group, told ABC the right of his party had not suggested a plausible alternative to McCarthy.

“You can’t beat somebody with nobody,” he said. “And right now you hear, ‘We’re just not going to vote for Kevin.’ Well, who then? Kevin deserves the opportunity. And he has done the hard work that was necessary to bring together the majority.”

Joyce said that though a moderate Republican with Democratic support “probably would be a perfect resolution … so we could start moving forward”, he did not “see it happening … I think the Democrats are going to vote for Democrats, Republicans will vote for Republicans. And I think, at the end of the day, Kevin will be the next speaker of the House.”

In the Senate, Democrats held control even before the Georgia runoff on Tuesday which will decide if they continue to rely on the vote of the vice-president, Kamala Harris, or by a 51-49 majority.

In his CNN interview, Jeffries was asked about comments by Mitch McConnell, the Republican leader in the Senate, who this week called Jeffries “a past election denier” over remarks about the 2016 presidential election, the legitimacy of which he questioned because of Russian interference, and Donald Trump.

Jeffries said: “If McConnell wants to lean into the fact that I’ve been critical of Trump’s presidency – the overwhelming majority of the world is critical of Trump’s presidency. That didn’t seem to make a lot of sense to me. But he’ll do what he does, and I want to stay focused on fighting for the people.”