WASHINGTON — When they met at the J. W. Marriott Hotel two blocks from the White House, Linda Singer, a former attorney general turned plaintiffs’ lawyer, approached Attorney General Gary King of New Mexico with an unusual proposition.

Ms. Singer wanted him to sue the owner of a nursing home in rural New Mexico that Mr. King had never heard of and Ms. Singer had never set foot in. She later presented him with a proposed lawsuit that did not cite any specific complaints about care. What she shared with him were numbers on staffing levels gleaned from records suggesting that residents were being mistreated there and at other facilities.

“Do you have 10 minutes at any point today?” Ms. Singer, who had served as attorney general in the District of Columbia, wrote to to Mr. King in a March 2012 email, to set up a meeting. “I finally got the numbers on the nursing home case and would love to discuss it with you briefly.”

“I’m in the lobby, near the reception desk,” Mr. King later replied, signing the message “GK.”

The casual nature of the exchange between the two Democrats, which was among thousands of pages of emails obtained by The New York Times, belied the enormous potential payoff for Ms. Singer’s firm if she could persuade Mr. King to hire her and use his state powers to investigate and sue, which he did.