Sharon Bialek, who worked for the NRA's education foundation from 1996-97, said she had contacted Cain in an attempt to land another job after her termination from the association.
Bialek's attorney, Gloria Allred, said Cain provided "his idea of a stimulus package."
The incident, which followed dinner at an Italian restaurant in Washington, D.C., was the first detailed allegation against Cain, who has been anonymously accused of sexual harassment by three other women who worked for the NRA in the late 1990s.
Bialek said she quickly extricated herself from Cain, saying, "This is not what I came here for." She related that Cain responded: "You want a job, right?"
Bialek said she didn't file a complaint at the NRA because she wasn't an employee at the time. She said she came forward now to "Give voice to those who have not come forward."
Referring to Cain, Bialek, a registered Republican, said: "I want you to come clean, admit you were inappropriate and then move forward. We need a leader who can set a good example."
Even before Bialek's press conference concluded, Cain's campaign dismissed what it called "false allegations" and "bogus attacks."
"Mr. Cain has never harassed anyone," the campaign statement said.
Polls released on Monday found that Cain remains one of the leading candidates for the GOP nomination and that many Republicans are convinced these charges are not serious.
A USA Today/Gallup poll released on Monday found a tight battle in the race for the Republican presidential nomination with two candidates tied atop of it.
Cain and former Gov. Mitt Romney of Massachusetts lead the pack with 21 percent each. Former U.S. House Speaker Newt Gingrich stands in third place with 12 percent, with Texas Gov. Rick Perry right behind him with 11 percent. The rest of the pack trails in the single digits.
The poll of of 1,054 Republicans and Republican-leaning independents was taken Nov. 2-6 and had a margin of error of +/- 4 percent.
Included in the USA Today/Gallup poll was a question on how Republicans view the charges leveled against Cain. Forty-seven percent of those surveyed believe the charges are probably false while 33 percent believe they are probably true.
The poll found that Republicans would turn against Cain if the allegations are proven true, with 53 percent saying they would never back a candidate who engaged in sexual harassment while 42 percent said they would be open to backing a candidate who did. Cain’s handling of the accusations garnered mixed marks. Forty-five percent of those surveyed think Cain has done a good job of handling the accusations while 36 percent say he has done a bad job with them.
The questions about harassment were asked of 850 Republicans and Republican-leaning independents; they were taken from Nov. 3-6 and also had a margin of error of +/- 4 percent.
Another poll released on Monday found that a majority of Republican voters are not concerned about the charges leveled against Cain. A new NBC News/Wall Street Journal poll found that 54 percent of Republicans said they are not concerned about the accusations while 13 percent said they have a “great deal” or “quite a bit” of concern about them.
Still, the NBC/Wall Street Journal poll found Cain is growing increasingly unpopular with voters across the board. In October, an NBC/Wall Street Journal poll had 24 percent of those surveyed with a positive view of Cain while 18 percent saw him in a negative light. The new poll has Cain upside down -- with 23 percent seeing him as positive while 35 percent see him as negative.
The NBC/Wall Street Journal poll of 1,000 adults was taken Nov. 2-5 and had a margin of error of +/- 3.1 percent -- with a margin of error of +/- 6.2 percent among Republican primary voters.