There’s a lot of chatter this morning about Mitt Romney’s claim last night that as Massachusetts Governor, he directed his administration to reach out to women’s groups to find female applicants for government jobs, resulting in them delivering him “binders” full of qualified women.

The tale isn’t quite how he told it, according to Liz Levin, the chair of a women’s group that was right in the middle of the story at the time.

In fact, Levin tells me, the groups initiated contact with him and urged him to hire more women — when he was still a candidate — and began creating the binders themselves on their own initiative before he took office. In fairness to Romney, she says, he did agree to work with them.

Asked yesterday by Candy Crowley about his thoughts on pay equity for women, Romney described his tenure in Massachusetts as follows:

I had the chance to pull together a cabinet and all the applicants seemed to be men. And I went to my staff, and I said, “How come all the people for these jobs are all men.” They said, “Well, these are the people that have the qualifications.” And I said, “Well, gosh, can’t we find some women that are also qualified?”

And so we took a concerted effort to go out and find women who had backgrounds that could be qualified to become members of our cabinet. I went to a number of women’s groups and said, “Can you help us find folks,” and they brought us whole binders full of women ... the University of New York in Albany did a survey of all 50 states, and concluded that mine had more women in senior leadership positions than any other state in America.

In 2002, when Romney was running for governor, Levin, who is now president of a management consulting company in Massachusetts, chaired a group called MassGAP. It was set up that year with the goal of increasing the number of women in government.

Levin, an Obama supporter, says MassGAP and other groups were the ones who first reached out to the gubernatorial candidates to get them to commit to hiring more women. She says Romney signed their pledge — as a candidate — and agreed to work with the groups.

At that time, the groups, on their own, were compiling lists of qualified female candidates for positions in state government, she says. After he was elected governor, in November of 2002, the groups took him a notebook full of those candidates.

“He did not initiate our project,” Levin says. “He was the recipient of a project we put together.”

Levin allows, however, that Romney was a “willing participant with us.” She says Romney designated Kerry Healey — now a top Romney adviser — as liaison to the groups.

At first, their collaboration brought results. “He did increase the percentage of women who were in state government, which we appreciated,” Levin says. However, by the end of his administration, she adds, the level of women in government had slid back down again.

Levin points to a University of Massachusetts study showing that by November of 2006, the level of women as a percentage of senior level positions had dropped to lower than it was when Romney took office.

Separately, David Bernstein has reported much the same thing, relying on multiple anonymous sources involved in the project at the time.