Segovia, the president of first division Union Espanola, is challenging incumbent Harold Mayne-Nicholls for the presidency and has considerable support from the top flight clubs including the "big three", champions Colo Colo, Universidad de Chile and Universidad Catolica.
Bielsa identifies strongly with Mayne-Nicholls, who appointed him in 2007 and was chairman of FIFA's inspection team that visited the countries bidding to host the 2018 and 2022 World Cups.
"I cannot and won't work in conjunction with Mr Segovia, it's impossible for me to do that," Argentine Bielsa told a news conference he called specifically to make his announcement.
Bielsa, who has also coached Argentina, signed a new contract until 2015 with Chile after steering them to the second round of the 2010 World Cup finals in South Africa where they lost to Brazil.
There had been growing speculation about Bielsa's future since the emergence of Segovia on a ticket to improve the standing of the clubs within the ANFP.
Bielsa has enjoyed complete autonomy over player selection during his tenure, something the clubs were not always happy about but he denied the existence of a clause in his contract permitting him to walk out if Mayne-Nicholls lost the election.
Bielsa, known for his long-winded explanations, took some time to make his point as he explained why he would feel sad at first if he had to resign but would feel better about it with time.
Segovia has said he would be glad for Bielsa to stay on but he has also talked about having "a plan B and a plan C".
Bielsa said he was against Segovia's idea to introduce a draft system for domestic player transfers as in Mexico, a plan also opposed by the Chilean players' union Sifup.
"It's true that players are bought and sold but the draft is an exaltation of the worst of this," said the 55-year-old Bielsa, who also coached Mexican clubs Atlas and America.
"I lived in Mexico for five years and if there is something that denigrates footballers it is the draft, said Bielsa, adding he was not surprised by the union's opposition to it.
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