In a phone call with the president before Saturday’s proposal announcement, Mr. McConnell encouraged Mr. Trump to extend the offer with temporary immigration protections as a way to reach out to Ms. Pelosi and other Democrats with a more appealing overture, according to a person familiar with the call who spoke on the condition of anonymity to discuss a private conversation.

Ms. Pelosi has argued that it is Mr. Trump’s intransigence — and Mr. McConnell’s refusal to allow a vote on House bills that would reopen the government but not fund a border wall, despite the Senate’s passage of such a measure last month — that is putting the nation at risk.

“The president’s insistence on the wall is a luxury the country can no longer afford,” Ms. Pelosi said on Thursday.

Senator Chuck Schumer of New York, the Democratic leader, suggested that Mr. McConnell was “way out to lunch” in trying to place responsibility for the shutdown on Ms. Pelosi. Her allies in the House say the speaker is hardly the sort to wilt under Mr. McConnell’s attacks.

“The bottom line is Pelosi is showing some leadership over here,” Representative Jim McGovern, Democrat of Massachusetts and chairman of the Rules Committee, said last week. “McConnell is behaving like a coward. He is afraid to take on Trump.”

As longtime party leaders, Ms. Pelosi and Mr. McConnell have history with each other, much of it not good given their divergent ideologies. Ms. Pelosi was a harsh critic of the 2017 tax law that Mr. McConnell championed. They have also clashed bitterly over the Affordable Care Act and the economic stimulus program enacted by Democrats under Ms. Pelosi’s stewardship during the first years of President Barack Obama’s administration despite Mr. McConnell rallying his party in near blanket opposition to the Democratic agenda.