That primary between Mr. Sanders and Mrs. Clinton left deep divisions in the party.

Democratic leaders worked assiduously to heal rifts and avoid a recurrence in 2020, in part by overhauling the party’s presidential nomination process. Specifically, they engaged in extensive outreach to Mr. Sanders’s fervent base of liberal supporters, who had come to distrust party leadership as beholden to major donors who favored centrist positions and supported Mrs. Clinton’s campaign. Some viewed the Center for American Progress, and its leader, Neera Tanden, as part of that cabal, working to stymie liberal activists and ideas.

The letter from Mr. Sanders, a self-described democratic socialist who is among the early front-runners in the 2020 Democratic field, threatens to undo a delicate rapprochement, and could presage another bitter primary battle.

The Center for American Progress, which is known as CAP, was founded in 2003 by John D. Podesta, a close ally of the Clintons, and is based in Washington. It has played an important role in Democratic politics, even as it is legally required to be independent from the party. It has been funded by major Democratic donors, like the financier George Soros, as well as entities with interests that do not always align with progressive politics, including health insurance companies, Walmart, big banks, defense contractors and foreign governments.

The group hosts forums featuring Democratic officials and candidates, including Mr. Sanders, while its action fund wages political advocacy campaigns targeting Republicans. It has also served as a Democratic administration-in-waiting, building teams and developing policies that President Barack Obama drew upon after his 2008 election. The group was expected to fill the same role for Mrs. Clinton had she won the White House in 2016.

Both Mr. Podesta, who was the chairman of Mrs. Clinton’s 2016 campaign, and Ms. Tanden, who advised it, were mentioned as candidates for White House chief of staff had Mrs. Clinton won. Ms. Tanden has privately disparaged some of Mr. Sanders’s supporters and advisers during the 2016 campaign. Among them was Faiz Shakir, who advised Mr. Sanders’ 2016 campaign and is his 2020 campaign manager. He worked at CAP for years, helping to launch and run ThinkProgress, which he defended in 2008 as independent from CAP, writing that the website “will oftentimes write items that are bolder, more strident, or more critical than what others here at the institution may be comfortable with.”