Le 10 Sport claims Les Verts are very keen on the Colombian international and scouted him in action against Young Boys on October 3 and again against Porto on October 24.
The 23-year-old has been linked with numerous clubs across Europe on the back of his displays for the Ibrox side with Aston Villa, Crystal Palace, Everton, Leicester, Newcastle, Real Betis, Borussia Dortmund, West Ham and many more clubs credited with an interest.
However, any move by Saint-Etienne would appear to hinge on the Ligue 1 side's outgoings.
Le 10 Sport believes the club would step up their interest in Morelos if Lois Diony or Robert Beric depart the Stade Geoffroy-Guichard in the near future.
The report also brands Rangers "very greedy" in terms of the price tag placed on the former HJK Helsinki frontman, with the Gers understood to be unwilling to accept anything less than £20 million for their star striker.
However, Saint-Etienne could revisit the transfer in the summer of 2020 if the price tag remains the same or thereabouts.