This year's planned intake of new staff triggered responses from more than 1100 potential recruits - the equivalent of 19 people chasing each trainee post.
There are a total of 58 apprenticeships up for grabs and, despite the rush for work, BAE Systems, which owns the warship yards at Scotstoun and Govan, has decided to extend the applications deadline for apprentice pipe fitters, fabricators and technical engineers until Monday, March 31.
Andrew Smith, HR Director at BAE's Naval Ships division, said: "Apprentices are a vital part of our continued investment in Glasgow so we've been very encouraged by the quality and diversity of applicants so far.
"These are valuable trades which can develop the full potential of both young men and women who want to work on some exciting ship build programmes, including the Queen Elizabeth Class aircraft carriers and Type 26 Global Combat Ships."
The Clyde yards have been tasked with building a number of hull sections for Britain's two new aircraft carriers - the biggest warships ever to be constructed in the UK - amid hopes that Glasgow will be chosen to build a fleet of new generation ships for the Royal Navy.
The engineering technical apprenticeship covers production and detail designs with a combination of on-the-job training and college classes.
Fabricators help build ships by constructing the frames and superstructure that form the skeleton of each vessel as well as the steel plates that cover them.
Pipe workers make and fit pipes the miles of pipe and cable needed to operate a modern warship.
Both Chloe Kirk and Taylor Thompson are trainee pipe fitters. Chloe, 20, from Greenock, Inverclyde, said: "It's great being an apprentice because you get to learn skills, while actually working and getting on-the-job experience. It's a really friendly atmosphere as I find everyone is here to help you learn."
Her 17-year-old colleague Taylor from Irvine, North Ayr- shire, added: "We've got a great team here as everyone is treated equal and our safety is really important. You get a really good induction into the business so I think this is a stable career with good prospects."
And 26-year-old Lyn Gordon from Greenock agrees. She now now works as an apprentice training officer and said: "When I applied for the apprenticeship, I was working in retail and didn't really see myself going anywhere."
BAE Systems is currently training 124 apprentices of all ages on the Clyde. It has one of the biggest apprenticeship schemes in Scotland. Potential trainees can apply online at www.baesystems.com/apprentices.