In the high-stakes negotiations between the Trump administration and Harvard University, the White House and a growing number of people at Harvard have at least one shared goal.

They want Penny Pritzker, the head of the university’s top governing board, out.

Ms. Pritzker holds a powerful post as the leader of the Harvard Corporation that in normal times is also a quiet one. The corporation is the equivalent of a board of trustees at any other academic institution and is mostly focused on fund-raising, strategy and picking the university’s president.

But for Harvard, these are far from normal times. The government has cut billions in federal funds to the school and tried to ban international students — a quarter of its enrollment — from attending. The school has sued the government twice in the last several months, even as it tries to negotiate an end to a conflict that has forced painful belt-tightening. As Harvard’s problems pile up, Ms. Pritzker’s leadership has been called into question.

On campus, prominent professors and donors are wondering whether she should go. And two Trump administration officials, who spoke on the condition of anonymity as delicate negotiations continue, say the hope is that the government’s pressure campaign on the school will lead to her ouster.

Ms. Pritzker, an heir to the Hyatt hotel fortune, has donated $100 million to Harvard for a new economics building. She has close ties with the Democratic Party and her family has a history of bad blood with President Trump. She has also faced criticism over her handling of the controversy surrounding Harvard’s former president, Claudine Gay, who resigned under pressure last year.

Now, some people tied to the school say that Ms. Pritzker has become a distraction that seems to be hurting rather than helping the university’s efforts to beat back an onslaught of Trump administration attacks. They suggested that she could be a concession whose resignation could help Harvard’s efforts to negotiate with the White House, if she would only agree to leave.