It is often said that Bing can appear in the most peculiar places and that’s true. 
In 1986 the Reagans gave a full-length interview with Barbara Walters when they were asked if they had co-star crush during their Hollywood careers, Nancy reveals that she “had a big, huge crush on Bing Crosby” and you can see that President Reagan seems genuinely surprised hearing it (considering that they must have known each other socially).
You can see the full interview here, Nancy’s contribution regarding Bing is on 8:08  https://www.youtube.com/watch?v=CLZpsNBMma8&t=626s
 
 
With Bing Crosby being one of Hollywood's most popular male stars, it's not hard to understand how an actress starting out in the early 50s would be a huge fan of him. Nancy Davis was born in 1921 and was a teenager when the crooner stole the heart of her generation, she literally grew up hearing his songs. She entered Hollywood by signing a 7-year contract with MGM and started appearing in films in 1949. Bing Crosby and the Reagans moved in the same circles for decades in Hollywood and California, so it's very likely they interacted on various occasions.
Both Reagans were huge fans of movies and regularly watched films at the White House cinema and their home cinema in Bel Air.