Taylor — who earns £2.29m a year — is the richest trade union leader in the world.
Many in the world of football think that, like Sepp Blatter — who was finally forced out as head of FIFA in 2015, when he was 79 — he has to go.
On TalkSport radio on Monday, November 19, footballer turned Hollywood film star Vinnie Jones said Taylor was a "dinosaur" and it was time for him to go.
The open letter has been signed by hundreds of footballers and former players, including ex-Premier League stars Chris Sutton, Robbie Savage and Ian Wright.
Taylor Has Run PFA Since Before Premier League Formed
A former winger with Bolton Wanderers and Birmingham City back in the 1960s, Taylor is now 73 years old and has been in charge of the PFA for 37 years.
The Times claimed sources within the PFA have questioned whether Taylor has stood for election in the last 10 years, as he is apparently supposed to under the union's rules.
On Friday, November 16, the chairman of the PFA, Ben Purkiss, called for an independent review into the functioning of the players' union, which was seen as the start of a challenge to Taylor's rule.
Purkiss, a 34-year-old defender who plays for Walsall in the third tier of English football, was elected chairman in November last year and is pushing for the union to modernise.
"I have a duty to act in the best interests of the PFA. Sometimes you have to make a stand for what is right. Football is rapidly evolving, players are rapidly evolving and the PFA needs to evolve too. Players past, present and future need a PFA for the modern player," said Purkiss.
PFA Criticised Over Dementia and Mental Health
Taylor, who retired in 1980, has been accused of not doing enough to investigate links between dementia and football injuries, especially head injuries.
Among those who criticised him over the weekend was Dawn Astle, whose father Jeff Astle, a former West Bromwich Albion and England player, died of a degenerative brain disease in 2002.
The Jeff Astle Foundation said there had been a "scandalous" failure to research whether repeated heading of heavy leather balls was linked to brain injuries and dementia in later life.
Purkiss has called for the PFA to play a greater role in helping players with mental health issues, an area in which the union under Taylor has been accused of failing.
Since the suicide of former Leeds and Wales star Gary Speed in 2011, a number of footballers have reported issues with depression, most recently former Manchester United star Michael Carrick.
Power Struggle in PFA
The Daily Mail claimed Taylor has reacted angrily to the challenge by Purkiss and has questioned his eligibility as chairman.
Taylor has written to the 92 PFA club delegates this week to adjourn the union's Annual General Meeting.
"There is a current issue which we are dealing with regarding the position of Ben Purkiss and his eligibility to remain as chairman and we have taken the decision to adjourn this meeting (AGM)," Taylor said in the letter.
It is thought Taylor feared the AGM would be used to oust him.
In 2008 Taylor agreed a deal with the News of the World whereby he was paid £700,000 in "hush money" and in return agreed to keep quiet about the paper's phone-hacking campaign.
Taylor had sued the paper after learning they had hacked his mobile phone voicemail messages.
During the litigation he discovered through the notorious "for Neville" email, that he was not a lone victim but phone hacking was widespread at the paper.
Three years later investigative journalist Nick Davies exposed the phone-hacking scandal and Taylor's deal.