De Boer only took over at Selhurst Park in the summer, but he already finds his job in serious jeopardy having seen his side lose all four of their opening Premier League games without scoring a goal.
Officials at the club reportedly held talks on Sunday night regarding the Dutchman's future, with the Burnley match considered by some to be a must-win if he was to stay in his role.
The Telegraph reports that the club will decide whether to sack De Boer within the next 48 hours, with newly-appointed sporting director Dougie Freedman understood to be on standby to take over.
Should De Boer be sacked then he would become the shortest-serving manager in the Premier League, having only taken over from Sam Allardyce in June.
Palace are the first team since Preston North End in 1924 to have lost all four of their opening top-flight matches without scoring a goal.