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.
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.
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.
