"I suspect it is to some degree damage limitation," said Nathan Sparkes, the chief executive of Hacked Off, quoted by the Guardian.
"In the case of The Sun, it is most likely because they are trying to avoid a full-blown court case in which lots of other details could emerge, which they don't want coming up."
According to the Guardian, Grant's case puts News UK in an awkward position. The company has already said the illegal activity only took place at News of the World, which closed in 2011.
Phone-hacking claims against The Sun - and subsequent settlements - have cost the news outlet millions so far.
Grant is not the only one looking to bring new phone-hacking claims forward.
Others include UK government minister Zac Goldsmith, his mother, Lady Annabel Goldsmith, football manager Alan Pardew, former cabinet minister Chris Huhne, Spice Girl Mel B, Atomic Kitten's Liz McClarnon, actors Gillian Anderson and Kate Winslet, boxer Joe Calzaghe and Steps singer Lisa Scott-Lee.
Grant had previously settled a phone-hacking case against the News of the World in 2012.
However, this new claim is specifically against The Sun.
Prince Harry is also currently going through the court process against the Sun.