The letter was addressed to Sir Keir, Home Secretary Shabana Mahmood in her role as chair of the NEC, and Labour Party General Secretary Hollie Ridley.
It said the decision to block Burnham had caused a “huge amount of anxiety and anger” amongst colleagues in the Parliamentary Labour Party and party members across the country.
Around half a dozen Labour peers also signed the letter, the BBC has been told.
Members of the NEC voted on Sunday morning to deny Burnham permission to stand in the contest.
Labour won the seat in 2024 with a 13,000 majority but Reform UK and the Greens are both expected to pose a greater challenge this time round.
Speaking to BBC Newsnight, former Labour cabinet minister Harriet Harman said her parliamentary colleagues should move on and focus on winning the by-election, instead of discussing the decision to block Burnham.