The Twilight Saga actor has been linked with the role of Finnick Odair - who is known by fans of the Suzanne Collins book trilogy for being the victor of the 65th Hunger Games - in the second movie in the film franchise, Catching Fire.
Sources told TMT that studio Lionsgate are keen to recruit the 26-year-old actor - who appears as vampire Edward Cullen in the Twilight movies - to star in the forthcoming motion picture, which will be directed by Francis Lawrence.
Pair worked together
Should Robert take on the role he will be reunited with Francis after the pair worked together on 2011 movie Water For Elephants.
Robert is due to star alongside his girlfriend Kristen Stewart in The Twilight Saga: Breaking Dawn - Part 2, the last movie in the franchise later this year, with production on Catching Fire set to begin this autumn ahead of its planned released date in November 2013.
Lionsgate recently signed Francis up to direct Catching Fire after The Hunger Games helmer Gary Ross revealed he would not be returning to the franchise.
Jennifer Lawrence, Josh Hutcherson and Liam Hemsworth all starred in The Hunger Games, which was released earlier this year.