Max McMullen, a Year 11 pupil at Beechen Cliff School in Bath, died in the city of Cordoba.
Local authorities say they are treating the incident as a tragic accident and say police are not investigating it as a crime at this stage.
The youngster fell to his death from the bathroom window at the block around 11.25pm on Saturday.
In a statement, Bath Rugby wrote: “It is with heartfelt sadness we share the news that Max McMullen, a young man who is in our Academy tragically passed away this weekend. Our thoughts are with Max’s family and friends at this deeply sad time.
“Max joined the Academy last season as an U15 having spent time previously in our Developing Player Programme here in Bath. He was a popular member of our Academy group, a young man that we saw develop both on and off the field during his time with us. He was a truly promising young player whose love for the game was clear for us all to see.
“We feel lucky as a club to have been able to support Max on his rugby journey in recent seasons, and this news is something that everyone in the Academy feels deep sadness about. As a club we will do all we can to support Max’s family and teammates both now and in the future.
It is with heartfelt sadness we share the news that Max McMullen, a young man who is in our Academy tragically passed away this weekend.— Bath Rugby (@bathrugby) October 22, 2019
Our thoughts are with Max’s family and friends at this deeply sad time.
➡️ https://t.co/9Qni1oebkx pic.twitter.com/MK2nJOKwES
“Max was a huge part of our academy and he will always be a part of our Club.”
Local news agencies reported that the boy had been trying to fix a broken blind when he died.
A source said it appeared the blind had got stuck as he tried to lower it after getting out of bed because a light coming through his window was stopping him from sleeping.