The House Republican leader, Kevin McCarthy, said on Saturday he had “reached out” to Democrats over Islamophobic comments made by one of his party, Lauren Boebert of Colorado, about the Minnesota Democrat Ilhan Omar.

Boebert apologised for the remarks, in which she likened one of the first Muslim women elected to Congress to a suicide bomber, on Friday, saying she wanted to meet Omar in person. Omar responded by condemning the remarks and calling for action from party leaders.

In a statement to CNN, McCarthy said: “I spoke with Leader [Steny] Hoyer today to help facilitate that meeting so that Congress can get back to talking to each other and working on the challenges facing the American people.”

McCarthy did not condemn Boebert’s remarks. He also faced criticism from within his own ranks, after another pro-Trump extremist, Marjorie Taylor Greene of Georgia, tweeted that she had “a good call” with McCarthy and liked “what he has planned ahead”.

Greene had criticised McCarthy, seeking to cast doubt on his ambitions to be speaker should as seems likely Republicans take back the House next year.

A Republican who spoke anonymously to CNN and was described as a moderate said McCarthy was “taking the middle of the conference for granted. McCarthy could have a bigger math problem [in the election for speaker] with the moderates”.

The anonymous moderate said his wing of the party – more of a rump, perhaps, given Donald Trump’s dominance – was upset about McCarthy’s embrace of extremists.

One such extremist, Paul Gosar of Arizona, was this month censured for tweeting a video which depicted him killing Alexandria Ocasio-Cortez of New York – like Omar a leading progressive and woman of colour in Congress – and threatening Joe Biden.

Gosar lost committee assignments. McCarthy said he would get them back under a Republican speakership and held out the same prospect to Greene, who was stripped of her committees in February for racist, antisemitic and generally incendiary behaviour.

McCarthy has faced calls from the right to punish Republicans who voted for the bipartisan infrastructure bill, as well as the 10 who voted to impeach Trump over the deadly Capitol riot.

Two who voted to impeach, Adam Kinzinger of Illinois and Anthony Gonzalez of Ohio, will retire next year. Primary challengers await the rest including Liz Cheney of Wyoming, a stringent conservative nonetheless split from the Trumpists over the Capitol attack.

On Saturday, Kinzinger criticised the minority leader’s call with Greene, writing: “Here is real strength, when Kevin McCarthy has to call a freshman begging for permission to stay in power. What has Kevin promised? The people deserve to know.”

He also said it had “been a while” since most “normal members … last talked to Kevin”.

The anonymous moderate who spoke to CNN said the party was on a “collision course” with itself, as their side “isn’t going to take this much longer”.

On Sunday, Asa Hutchinson, the governor of Arkansas who is seen by some as a possible presidential nominee from the more moderate side of the party, told CNN’s State of the Union McCarthy should have condemned Boebert.

“Even in our own caucus, our own members, if they go the wrong direction, I mean, it has to be called out,” Hutchinson said. “It has to be dealt with particularly whenever it is breaching the civility, whenever it is crossing the line in terms of violence or increasing divides in our country.”

Earlier this week, Jackie Speier, a senior Democrat from California, told the Guardian McCarthy had “a number of radical extremists in his caucus that are very effective communicators to the right fringe, and he can’t really rein them in because reining them in means they will attack him.

“So they have become the face of the House Republicans. You might as well put a brass ring in Kevin McCarthy’s nose because they’re pulling him around.”