Durham City Rugby Club, which oversees the team, said in a statement the pair suffered "non-rugby related breathing problems".
In a statement, Durham City Rugby Football Club said it was with "great sadness" that they announced the death of 26-year-old Tom Baty on Tuesday afternoon.
British rugby is in shock following the death of a club player while touring Sri Lanka.
Durham City RFC president Richard Wilkinson said: "On behalf of all members and friends of the DCRFC family we would like to extend our honest condolences to the family of Thomas Howard".
She added: "We are assisting the family of a British man who has been hospitalized in Sri Lanka, and are in contact with the Sri Lankan medical services". Our thoughts and prayers are with you all. Mr Howard's death was announced on Monday.
In a statement released on Tuesday, the club requested that the privacy of their families be "respected and upheld at this hard time".
Mom hears screams, sees bear drag off her 5 year-old daughter
The girl's mother heard her screaming, went outside and started screaming at the bear, causing it to drop the girl. The area is nestled between the Gunnison and Colorado rivers, which provide corridors for migrating wildlife.
England rugby player Tamara Taylor was among those offering their sympathies on social media.
"Thoughts and prayers at this hard time".
However, after returning to the team hotel, two players had developed breathing difficulties and the hotel management had admitted them to a leading private hospital in Colombo.
After the match, they had visited the club and team members returned to their hotel in smaller groups by early Sunday.
Another British Rugby player who was admitted along with Bett Thomas Reid passed away that very night with the same symptoms.
One local report said the family of the dead man, aged 26, would travel to Sri Lanka from Northern Ireland.
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