Mr. McCormick, then the chief executive of the world’s largest hedge fund, had an edge in pitching Mr. Trump: His wife, Dina Powell McCormick, had been a senior national security official in the Trump White House, and she accompanied him to the meeting at Mar-a-Lago.

As Mr. McCormick and his wife, now a top Goldman Sachs executive, made their case, the topic soon turned to electability and Dr. Oz’s Turkish American heritage, which has since become a central point of contention in the campaign. At one point, Ms. Powell McCormick, an Egyptian-born Coptic Christian who is fluent in Arabic, pulled out a picture that showed Dr. Oz alongside others wearing Muslim head coverings, according to four people briefed in detail on the exchange, which has not previously been reported.

The people briefed on the conversation said Ms. Powell McCormick told Mr. Trump that the fact that Dr. Oz was Muslim would be a political liability in parts of Pennsylvania.