The 25-year-old, recently dropped from the Great Britain team, alleged that Sutton made sexist comments and told her to "go and have a baby".
"I want to change the culture and their treatment of women," she said.
"I hope by shining a light and sharing my experiences, the relevant people can investigate and make changes."
UK Sport, which funds British Cycling's elite performance programmes, said it had been in contact with British Cycling to ensure the allegations were investigated.
"We take allegations of sexism extremely seriously,"  it said. "As a recipient of funding, British Cycling should assure us this will be dealt with fairly, swiftly and transparently.
"We expect the highest levels of professionalism from athletes and staff and have zero tolerance to sexist behaviour."
Varnish has been been invited to meet the equalities officer of British Cycling to discuss her concerns.
A British Cycling spokesman said: "A gold medal is valued by us, no matter who wins it and we are equally proud of all our Olympic and world champions."
He added that medals won by the Great Britain cycling team are "testament firstly to the dedication and talent of our riders" but also a "tribute to a significant investment in passion, commitment and resources by our performance staff".
Sutton, 58, denies any wrongdoing and says British Cycling did not renew her contract because her times had slowed over the past three years.
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Varnish added: "I remain open to sharing my experiences with both British Cycling and/or UK Sport, and will happily engage with any investigations into the comments that Shane Sutton has made to me, and other riders."
The Worcestershire-born rider also said she would like to compete for Britain again, adding that she was not too old and "not a waste of UK Sport's money".
Earlier, Olympic champions Victoria Pendleton and Nicole Cooke both criticised British Cycling following Varnish's claims.
"I know exactly how miserable they made me," two-time Olympic gold medallist Pendleton told the Daily Telegraph.
"I never really felt I had the same respect as my male team-mates."
Cooke, the 2008 Olympic road race champion, told the Guardian: "I have my own experiences of Shane and sympathise with Jess.
"Speak out and your dreams will be destroyed and years of hard work wasted. Or put up with it and hope."
"I spoke out from the age of 19 and I know what happens."
Olympic gold medallist Joanna Rowsell Shand said on Monday that she was "surprised" at Varnish's claims of sexism.

Keepers at Drusillas Park in Alfriston could not work out why Sofia and male Tupee had failed to bond since being introduced to each other in 2011.
Head keeper Mark Kenward said it was not easy to establish the sex of a sloth and medical records had shown Sofia was female when she arrived.
"We are going to have to find a more suitable name for Sofia," he said.
It is likely that either Tupee or Sofia will now be re-homed to another zoo, and another female introduced.
Mr Kenward said: "We had never looked after sloths before and were relying on the information provided.
"Unfortunately we cannot keep our two boys together and it would also be a waste of their breeding potential."
After enlisting the help of Bristol Zoo, Mr Kenward said it all made "perfect sense" given the problems they had been experiencing with the animals.
Jess Varnish says she spoke out against British Cycling technical director Shane Sutton with the hope of changing attitudes at the organisation.
[NEXT_CONCEPT]
A pair of sloths at an East Sussex zoo failed to breed after it emerged that they were both in fact males.