She also spoke by phone with the foundation’s directors to explain what happened with Mr. Lynn and his initiative. Most members of the board either declined to comment or could not be reached.

But some of New America’s donors were unsettled by the news, according to an official at one foundation that donates to the think tank. “We were concerned because you want to let the grantees do their work without worrying about how it impacts the funders,” said the foundation official.

Another liberal think tank, the Center for American Progress, which has faced occasional scrutiny over its positions and its donors’ interests, highlighted the New America situation as a cautionary tale in an internal email.

CAP’s president, Neera Tanden, wrote to her staff on Wednesday afternoon that the episode serves as “a good reminder that every institution’s ability to impact the national debate is based on trust.” Describing CAP’s guidelines as dictating that “the financial interests of any donor shall have no influence over our policy work or the policy positions we support,” Ms. Tanden wrote, “We never should come close to the line on these issues.”

Ms. Slaughter, in an interview on Thursday night, said her conflict with Mr. Lynn had nothing to do with his ongoing tough criticism of Google. Rather, she said it stemmed from his repeated violations of the think tank’s “strong implicit norms about providing a heads-up when you are doing something that could have an impact on the funding for your fellow directors.”

Ms. Slaughter, who served as the dean of Princeton University’s Woodrow Wilson School of Public and International Affairs before joining the Clinton State Department in 2009, said, “I spent my life in the academy. Of course, I believe in the independence of research, but you have to give people a heads up when you are going to criticize an institution upon which others rely for funding.” The New America Foundation paid her $535,000 in salary and other compensation in 2015, according to the organization’s tax filings.