Sports minister Hassan Wario launched his stinging criticism during a Kenyan television show on which the acting president of Athletics Kenya, the track federation, was also appearing.
The sport in Kenya is in a mess, with more than 40 athletes failing doping tests since 2012 and four high-ranking federation officials under investigation for doping cover-ups and other alleged wrongdoing.
"Probably the way out of this is a very free and fair election so that we get a new team of people on board and we engage with them," Wario said on the TV show, challenging AK's acting head Jackson Tuwei, his fellow guest. "Kenyans have lost faith in what is going on in that federation as it is now."
Wario also said he'd met recently with Coe, and the world governing body's patience with Kenyan athletics officials was running out.
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Three of Athletics Kenya's most senior officials have been suspended by the International Association of Athletics Federations, pending investigations by its ethics committee over alleged doping cover-ups.
AK President Isaiah Kiplagat, vice president David Okeyo and chief executive Isaac Mwangi are all under investigation, as is the federation's former treasurer.
Mwangi was placed under investigation this week after two athletes who failed doping tests alleged in an interview with The Associated Press that he tried to extort bribes totaling $48,000 from them in exchange for organizing lenient bans.
allegations that they embezzled money meant for the federation from sponsor Nike. Kiplagat, Okeyo and former treasurer Joseph Kinyua, who was the manager of Kenya's team at last year's world championships, are accused of embezzling around $700,000.
But Wario continued his attack: "One of the major problems with federations is people staying there for 30 years or 40 years without putting proper structures, with very little transparency," he said. "For instance, Nike deals don't pass through any government agency. We don't know. It's shrouded in secrecy. Everything is done behind us."
Earlier, Wario said new legislation to make doping a criminal offense will be presented to Kenya's parliament on Monday as the country scrambles to avoid being declared non-compliant with world anti-doping rules. The proposed legislation was now ready to go to lawmakers for approval, he said.
That could lead to a ban for its track and field athletes from international competition, including the Rio de Janeiro Olympics. Coe previously said that the IAAF would consider banning Kenya, like it did with Russia, if the country was in breach of anti-doping rules.
Kenya's government must also provide proper funding for the new national anti-doping agency. The Anti-Doping Agency of Kenya received $3 million of the $5 million the government approved for its setup costs only 10 days ago, said its chief executive, Japhter Rugut.
WADA has been critical of Kenyan authorities, who have had nearly four years to act after serious problems of doping among its runners were uncovered in 2012.