Initial investigations showed the two victims from the Durham City Rugby Football Club, along with their team mates, had played a friendly with the Ceylonese Rugby and Football Club on Saturday.
Durham City Rugby Club, which oversees the team, said in a statement the pair suffered "non-rugby related breathing problems".
A police statement said: "A 26-year-old player died while undergoing treatment at the hospital on Sunday afternoon, while a 27-year-old player is undergoing treatment".
"Both players and Clems Pirates hold a special place in our rugby family, and our continued support is unconditional", said Richard Wilkinson, president of DCRFC.
In a statement released on Tuesday, the club requested that the privacy of their families be "respected and upheld at this hard time".
A spokesman for The Foreign and Commonwealth Office said: 'Our staff are supporting the family of a British man following his death in Sri Lanka, and are in contact with the Sri Lankan hospital services.
The Sri Lankan police force was not immediately available for comment when contacted by CNN.
Clems Pirates are described as a veterans' team who raise money for charities in the United Kingdom and overseas.
It added: 'As a mark of respect to the deceased players, Clems Pirates will go ahead with their planned charity work and rugby kit donation in Galle before the end of the tour'.
"Such very bad news", posted Aviva Premiership club Newcastle Falcons on its official Twitter account.
"Our thoughts and prayers are with you all", he said.
"We would like to extend our honest condolences to the Baty family".
England rugby player Tamara Taylor was among those offering their sympathies on social media.