Senator Mark Warner, Democrat of Virginia, is working to convene Democratic senators next week to discuss a path forward after President Biden’s disastrous debate performance, and to discuss their concerns about him remaining as the nominee, according to five people with direct knowledge about the effort.

The push by Mr. Warner reflects a mounting sense of panic among some Democrats in the Senate about Mr. Biden’s viability to continue in the presidential race, and growing frustration among senators that the president and those around him have not communicated directly with them about how they plan to address such concerns.

The people insisted on anonymity to discuss Mr. Warner’s efforts, and a spokeswoman for the senator did not respond to requests for comment. His outreach efforts were reported earlier by The Washington Post.

While some House Democrats have been outspoken in their harsh assessments of Mr. Biden’s performance last week — and three have called on him to end his candidacy — most senators have so far been quieter about their concerns.

Mr. Warner, a centrist former governor who himself weighed a presidential run in 2008, has been reaching out to colleagues since the night of the debate last week to express anguish about Mr. Biden’s performance and a sense of urgency for Democrats to figure out what to do about it. As chairman of the Intelligence Committee, he is a trusted voice in the Senate and has cultivated a reputation for bipartisanship.

While his conversations with Democratic senators have conveyed his dismay about the situation, Mr. Warner, according to one person who spoke with him on Friday, was not trying to organize an effort to persuade the president to withdraw from the race but was instead working to facilitate a discussion about the right way to proceed.

Mr. Biden told reporters on Friday that he had spoken to at least 20 members of Congress, and “they’re telling me to stay in the race.”

When a reporter said Mr. Warner was trying to get him to step aside, Mr. Biden responded that the senator was “the only one considering that. No one else has called on me to do that.”

Democratic Representatives Lloyd Doggett of Texas, Raúl Grijalva of Arizona and Seth Moulton of Massachusetts have all called on him to do so this week.

Senators are scheduled to return to Washington on Monday after a weeklong recess.

Luke Broadwater and Carl Hulse contributed reporting.