Jack, 20, started the course towards the end of the year and is due to finish this autumn.
He has decided against pursuing his dad's steps in the Michelin chef industry and will be spending the next few months at the Royal Marines' Commando Training Centre at Lympstone, Devon.
Despite his dad's celebrity status, Jack will be earning £18k as a recruit.
A source at the training centre in Devon claimed Jack’s fitting in well.
"First and foremost, he is just another marine recruit to us, a 'bod'. There’s no nepotism here and no favouritism," a source told the Mail.
"He’s getting his head down and delivering the goods so far. He seems fit and is well liked.
"Not surprisingly, there’s a lot of good-natured banter when Jack and his mates sit down to eat and when they’re on a field exercise about who should prepare the rations."
According to the Daily Record, TV chef Ramsay, 53, said: "He [Jack] is obsessed with Bear Gryulls and the Marines , and going in and becoming an officer. That's a real job."