According to the BBC Sport, the Championship side has approached the Football Association of Wales regarding the appointment of the 46-year-old as Scott Parker’s successor, with contract discussions now underway.
Although no final agreement has been reached, talks have intensified in recent days and significant progress is believed to have been made.
Bellamy had initially been expected to remain with Wales through to Euro 2028, which will be hosted across the UK and Ireland.
However, the former Manchester City, Liverpool and West Ham winger is now seriously considering a return to club management.
Bellamy has been in charge of Wales since 2024 but was unable to guide the nation to the 2026 World Cup.
A move to Burnley would mark a return to Turf Moor, where he previously served as Vincent Kompany’s assistant.