But Mr. Johnson, who is seeking to reach 15 percent support in national polls to qualify for the presidential debates, also sharply criticized Mrs. Clinton on Tuesday. He described the activities of the Clinton Foundation as a “pay to play” scheme, whereby foundation officials sought donations from people seeking access to the government.

“It’s so blatant and it’s so obvious,” Mr. Johnson said.

Mr. Johnson has strained to gain ground in the general election as an alternative to both major-party candidates, despite the low regard in which many voters hold Mr. Trump and Mrs. Clinton. Should Mr. Johnson fail to qualify for the debates, his odds of breaking out as a third force in the presidential race would probably be greatly diminished.

Mr. Johnson said his campaign might air national cable advertisements after Labor Day to boost his name recognition. But he said fund-raising would determine whether such a move would be possible. He also listed a number of states he and Mr. Weld saw as winnable, including Utah, Alaska and the Dakotas.

Though he was elected governor as a Republican, Mr. Johnson has collected endorsements from only a handful of the many mainstream Republicans who have withheld their backing from Mr. Trump.

Mr. Johnson said he was not optimistic that he could win endorsements from major anti-Trump Republicans, like Jeb Bush and Mitt Romney. The pressures of party loyalty, he said, remain strong.