Mr. Trump recently softened his position on immigration, forgoing his calls for mass deportation in favor of a focus on “criminal aliens.” Mr. Pence was also one of the first members of the campaign to publicly acknowledge that President Obama was born in the United States, paving the way for Mr. Trump to finally end his false conspiracy theory about Mr. Obama’s birthplace last month.

During the vice-presidential debate this week, Mr. Pence brushed off many of Mr. Trump’s startling comments from the campaign, disregarding some as reflecting a lack of political polish and denying that others were ever said.

But critics of Mr. Trump are not letting his campaign off the hook for the Muslim ban so easily.

“Governor Pence’s flagrant attempts to mislead voters on his running mate’s positions aren’t fooling anyone,” said Zara Rahim, a spokeswoman for the Clinton campaign. “Not only has Trump proposed an unconstitutional immigration ban on an entire religion, but he’s suggested creating a database that tracks Muslims in this country.”

She added, “Pence has not disavowed anything, he’s just lied to the American people once again.”

Nihad Awad, the executive director of the Council on American-Islamic Relations, said that Mr. Trump could not simply turn the page on the Muslim ban.

“Whatever the Trump campaign claims is the current version of its Muslim ban, the original absolutist language of a ‘total and complete shutdown of Muslims entering the United States’ — along with other bigoted statements — reflect a systematic and toxic use of Islamophobia that has had a tremendously harmful impact on the lives of ordinary American Muslims and on the unity of our nation,” Mr. Awad said.

Despite efforts to quietly backtrack on the proposal, the Trump campaign has continued to face questions about the ban because the news release from last December announcing the proposal has not been removed from Mr. Trump’s website.