Hudson, 39, and King, 79, released the original song "Here I Am (Singing My Way Home)."
Hudson and King co-wrote "Here I Am," which will appear on the soundtrack for Hudson's film Respect. Hudson plays Franklin, who died at age 76 in August 2018, in the new movie.
King co-wrote Franklin's hit song "(You Make Me Feel Like) A Natural Woman" with Gerry Goffin.
Respect follows Franklin's rise to fame as an international music superstar known as the Queen of Soul. The film is directed by Liesl Tommy and co-stars Forest Whitaker, Marlon Wayans and Audra McDonald.
Hudson shared a trailer for the film on Good Morning America in May. Hudson also recalled how Franklin chose her to portray her in the film following Hudson's role in Dreamgirls.
"For her to choose me, I can't even begin to put it into words. I do not take it lightly and I just think it's a dream come true. I don't want to sound cheesy but it is," Hudson said.
Respect opens in theaters Aug. 13.