If the story he told you was that Harry "White Fang" Cohn wanted her himself- he had the services of each and every of the girls under contract for a week apiece if they wanted to be signed with Columbia, as Marilyn Monroe learned in 1948- it's not *quite* that simple. On one occasion when Cohn demanded her services, she told him she was in love with Sinatra, and they both were blacklisted from the lot.
Chicago mob boss Tony Accardo owned a part of Monroe's contract then and had Chicago's west coast mob ambassador to Hollywood Johnny Accardo discuss the matter with Cohn, as the two were particularly close associates. That was about the time Cohn was producing the film *From Here to Eternity* the part which Frank Sinatra desperately needed to salvage his tanking career. If you've seen the movie *The Godfather* just recall the part of the film involving the producer and the horse's head....
Cohn took both Monroe and Sinatra back into the Columbia fold, and Sinatra got his part as the Italian soldier Maggio in *From Here to Eternity*; Monroe, who Cohn contemptiously dismissed as a no-talent tramp, got an abortion about that time; the kid was probably Cohn's. Monroe's connection was through the Chicago Outfit; Sinatra's via Frank Entratta, who then ran the Sands Casino in Las Vegas for New York Mafia boss Frank Costello. And not only did Sammy Davis Junior's trip into the desert from which his return depended on his answer inflluence his decision regarding Kim Novak, so too did the advice he got from Sinatra. Though Cohn may well have had his fun with Novak as his studio droit de segnuir, his interest was much more in reminding Novak and Monroe, and the others like them, just who the boss was, and why it was a good idea to keep him happy.
Interestingly, Rita Hayworth, Kim Novak and Sinatra all worked together on the 1957 film of the musical Pal Joey. That must have been interesting....