Their hit, La Bicicleta, won last year's Grammy Latino.
Cuban musician Livan Rafael Castellanos claims that Shakira and Vives copied a line and part of the chorus of his 1997 song Yo Te Quiero Tanto.
The complaint was filed at a court in Spain, where Shakira lives.
A spokesman for the court said the case centred around an intellectual property problem and focuses mainly on the melody and the lyrics of the choruses of both songs.
In La Bicicleta (or The Bicycle) Shakira and Vives sing "que te sueno y que te quiero tanto" (I dream of you and love you so much).
The Cuban artist, known as Livam, sings "yo te quiero, yo te quiero tanto" (I love you, I love you so much).
El Mundo newspaper in Spain reported that the two sides had a meeting in October but they failed to reach a settlement.
AFP news agency says Shakira's team has refused to comment on the case.
(Picture: La Bicicleta was the first song written together by the two Colombian pop stars; BBC)