Mr. Obama addressed the group four nights before Tuesday’s nominating contests, in which Mrs. Clinton was heavily favored. As it happened, Mrs. Clinton won at least four of the five states that voted — Missouri has yet to be called — further padding her delegates lead.

Mr. Obama indicated that he knew some people were not “excited” by Mrs. Clinton’s candidacy, a White House official confirmed.

But, while he stressed that he was not endorsing either candidate, and that both would make good presidents, Mr. Obama went on to lavish praise on Mrs. Clinton, describing her as smart, tough and experienced, and said that she would continue the work of his administration. Mr. Sanders has publicly criticized Mr. Obama on certain policies and has called for a “political revolution.”

Mr. Obama said that he understood the appeal to voters of a candidate who is authentic, the official said. But he also reminded the Texas donors in the room that Mr. Bush was considered authentic when he was running for president, suggesting that being authentic did not necessarily translate into being a good president, in his view.

Mr. Trump’s admirers have often praised him for his authenticity and blunt style, contrasting it with Mrs. Clinton’s more cautious approach.