The Lancashire club extended their winless run across all competitions to nine matches on Saturday afternoon, losing 3-0 at Selhurst Park against Crystal Palace.
After Burnley captain Josh Brownhill received his marching orders in the first half, Oliver Glasner's Eagles always looked destined for maximum points, with Chris Richards, Jordan Ayew and Jean-Philippe Mateta all breaching the net of James Trafford in the second period.
Defeat in the capital leaves the Clarets 19th in the Premier League standings with just 13 points from 26 top-flight matches, 11 points behind 17th-placed Nottingham Forest.
According to The Athletic, Burnley have no current plans to dismiss manager Kompany following a difficult campaign so far on their return to the Premier League.
Manchester City legend Kompany will allegedly continue to receive the backing of the Clarets' hierarchy heading into the business end of the term.
The same cannot be said for sections of the club's fanbase, though, who made their discontent towards the 37-year-old known following the defeat at Palace on the weekend.
Following a maiden managerial position at Belgian Pro League outfit Anderlecht, Kompany joined Burnley in the summer of 2022 and guided the club to the Championship title during his first season in charge.
Amassing 101 points and losing just three of their 46 second-tier matches, the 37-year-old was able to return the Clarets to the Premier League at the first attempt.
Unwilling to depart from a playing style that served his side handsomely in the Championship, Kompany has had little joy during
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