The club issued a statement which said: "The Howard and Baty families have requested that their right to privacy is respected".
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.
The club previously said that in both cases the breathing problems were non-rugby related.More news: Netta Barzilai wins 2018 Eurovision Song Contest for Israel
The 22-strong party from the club arrived in Sri Lanka last Thursday and played a friendly against a local team two days later.
A spokesman for the Foreign Office said: "We are assisting the family of a British man who has been hospitalised in Sri Lanka, and are in contact with the Sri Lankan medical services...."
In a statement released on Tuesday, the club requested that the privacy of their families be "respected and upheld at this hard time". "Our thoughts are with you all and we are here to help in any way we can".
People have been paying tribute to Mr Howard on the club's Facebook page.More news: Found: A Rapidly Expanding Supermassive Black Hole
Former England women's captain Tamara Taylor called it "unbelievably tragic news", sending "love to all at Clems Pirates and Durham City RFC".
They visited a nightclub in Colombo after a match against a local team and complained of breathing difficulties after returning to their hotel.
A spokesperson from the Foreign and Commonwealth Office (FCO) said earlier: "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".More news: Project Maven: Nearly a dozen Google employees have reportedly quit in protest
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