Among those who said they won't run are former U.S. Rep. Pat Williams, former Gov. Brian Schweitzer, state Auditor Monica Lindeen and Superintendent of Public Instruction Denise Juneau.
Other names floated by Democrats were Zeno Baucus, the former senator's son, and former state legislator and abortion rights leader Nancy Keenan.
But Zeno Baucus told The Associated Press on Friday he was committed to his job as an assistant U.S. Attorney in Helena and not interested in the Senate candidacy, and Williams said Keenen told him "absolutely not."
Despite the apparent long odds any candidate will face against Daines' well-financed campaign, Williams said he's hopeful an unexpected candidate will emerge to stir up voters who might be otherwise disinterested.
"Democrats need to push the reset button, and if they do that properly, they can beat Steve Daines," he said. "It might be a candidate that Democrats are talking about but is new to the public."
He said either Wilmer or Curtis would fit that description. Both said Friday that they're interested in running against Daines.
A spokeswoman for U.S. House candidate John Lewis said he's been approached to enter the Senate contest but remains focused on the House.
A new candidate must be selected before Aug. 20. Democrats have arranged for a nominating convention on Aug. 16 at the Lewis and Clark County Fairgrounds in Helena. It will include leaders from county party committees, along with federal and statewide elected officials, and the party's executive board.
Walsh, a former lieutenant governor and decorated veteran who served in Iraq, was appointed to the Senate in February by Gov. Steve Bullock. He bowed out of the campaign Thursday after his campaign was derailed by a New York Times report that revealed the extensive use of unattributed material in a 2007 paper Walsh submitted to earn a master's degree from the U.S. Army War College.
Walsh was in Montana Friday, but had no public events scheduled and was not available for interviews, according to his Senate and campaign offices. His office sent out a statement Friday afternoon urging a clear mission and end goal regarding U.S. airstrikes in Iraq.
Walsh's first scheduled appearances will be Tuesday, when he plans a series of roundtable discussions about Social Security and Medicare in Missoula, Great Falls and Lewistown, said Walsh's Senate spokeswoman, Andrea Helling.
The Senate is in recess until Sept 8.
"He will be traveling around Montana, meeting with Montanans about issues Montanans care about," Helling said.
Two-term former Gov. Brian Schweitzer, once considered a shoo-in for the Senate seat but who has insisted he's not interested in running, said Friday that the field of potential candidates is shifting rapidly. He indicated there was no clear favorite at this point.
"It's too dynamic. There's people every hour saying 'I'm out,' and every hour new names coming," he said.
Schweitzer, who serves as chairman of Stillwater Mining Co., the state's largest company by revenue, added that his decision to stay out was final, but he suggested he could have won if he entered.
"It was a final thing 14 months ago," he said, referring to Baucus' retirement announcement last year. "Fourteen months ago, had I run, Steve Daines would still be in the House, and I would be running against (former Republican state Sen.) Corey Stapleton, 20 points ahead."
Brown reported from Billings.