The Clarets had identified Bellamy as the successor to Scott Parker and held talks with the Football Association of Wales.
"The Association and (FAW chief executive) Noel Mooney know that Bellamy is looking at other jobs and has had his head turned by the links to Burnley," said Roberts to BBC.
"The big question now is whether they keep him on as national team manager.
"He's lost a lot of love and faith among the fans and I would think he's burnt a lot of bridges."
The breakdown is not believed to be related to compensation, with discussions over Bellamy's backroom staff reportedly proving a sticking point.
Bellamy previously served as Vincent Kompany's assistant at Burnley between 2022 and 2024 and remains under contract as Wales manager until 2028.