British Cycling's Shane Sutton said the decision not to renew her contract "was based on a projection of medal competitiveness at Olympic level".
Varnish, 25, missed out on Rio after she and Katy Marchant finished fifth at last month's World Championships.
Afterwards she strongly criticised decisions made by British Cycling.
Media playback is not supported on this device
"I feel sick," Varnish told BBC Sport at the time.
"We have been basically playing catch-up for two years after decisions that were [made] above us, and bad luck."
But British Cycling's technical director Sutton insisted the decision to drop Varnish had been made purely on performance grounds.
"Since her gold medal at the London World Cup in 2012, Jess had not made the necessary gains in performance compared to the rest of the world," he said.
Speaking to the Daily Telegraph, he added: "There is no point carrying on and wasting UK Sport's money on someone who is not going to medal going forward."
Britain's two places in the individual sprint at the Rio Olympics are now likely to be filled by Marchant and Becky James.
Varnish raced in the team sprint with Victoria Pendleton at London 2012, breaking the world record in qualifying before being disqualified in the semi-finals.
Varnish subsequently won a bronze alongside James at the 2014 World Championships, as well as two individual bronzes for England at the Commonwealth Games in Glasgow that year.
Sprinter Jess Varnish has been dropped from the Great Britain team after failing to qualify for the 2016 Olympics.