The Clarets drop back into the Championship for the second time in three seasons having won just four of 34 league games, taking a single point from their final eight matches.
Bellamy is viewed as a serious candidate at Turf Moor but convincing him will not be straightforward, reports BBC Sport. He has history with Burnley, having served as a coach under Vincent Kompany and been among the candidates for the top job before Parker was appointed in 2024. But his situation has changed significantly since then.
He is said to be enjoying his time with Wales and is focused on leading the nation to a home nations European Championship on home soil in 2028 — a project that gives him a clear long-term purpose and a timeline beyond the immediate challenge of rebuilding a relegated squad.
Bellamy also holds reservations about the current state of the Burnley playing squad and the scale of reconstruction required this summer, with a significant number of departures expected.
No formal approach has been made, and Burnley would need to pay a release clause of at least £700,000 to the Football Association of Wales to open the door to any negotiation.
Whether Burnley can offer a compelling enough project to pull Bellamy away from international football — and from a team building towards a major home tournament — remains the central question in what looks a difficult pursuit.