Harvard University became the leader of academia’s resistance to the Trump administration — and soaked in acclaim from the White House’s critics — when it refused a roster of intrusive demands and took the government to court last month.

Legal experts saw a strong case, built by a team of elite conservative lawyers, to win back billions of research dollars that the government had stripped away. Supporters cheered on Harvard’s unusually sharp public tone.

“Congratulations to Harvard for refusing to relinquish its constitutional rights to Trump’s authoritarianism,” Senator Bernie Sanders of Vermont wrote on social media last month.

But behind the scenes, several senior officials at Harvard and on its top governing board believe that the university is confronting a crisis that could last until President Trump is out of power, according to three people involved in the discussions. Even if Harvard’s legal case is successful, these officials say, the school will still face enormous troubles that may force the nation’s oldest and wealthiest university to rethink its identity and scale.

Any outcome seems likely to lead to significant cuts to Harvard’s research and work force and undermine its pre-eminence for years. Without its sprawling research apparatus, there is a fear that it could become more like a small, teaching-focused liberal arts college.

University leaders believe the only clear options are either working with Mr. Trump or somehow securing huge sums of money quickly, perhaps from private donors, the three people said. They spoke on the condition of anonymity because they had not been authorized to discuss school officials’ private deliberations.