Harvard representatives sought a meeting after other higher education leaders expressed hope that it — on behalf of academia — would re-engage with the administration. And Harvard’s outreach came after Education Secretary Linda McMahon publicly raised the prospect of negotiations with a university she routinely criticized. Harvard officials sensed an opening and suggested a briefing on steps the school has taken in recent years, two of the people said.

It is unclear how close both sides are to a potential deal and the exact terms any final agreement would entail. In a post on Truth Social, Mr. Trump said it was “very possible that a Deal will be announced over the next week or so.”

Two people briefed on the discussions said it was highly unlikely a deal would be reached in the next week.

Harvard has been widely praised by Democrats, academics, its alumni and democracy advocates for fighting the Trump administration. But top Harvard officials, according to two people briefed on the matter, have become increasingly convinced in recent weeks that the school has little choice but to try to strike a deal with the White House.

The Harvard officials believe that if the university remains at odds with the administration that it is likely to become far smaller and less ambitious as Mr. Trump tries to keep pummeling it with funding cuts, federal investigations and limits on visas for international students.