The appointment has not been confirmed, but the Australian politician is widely expected to take up a role on Britain's re-launched Board of Trade.
Rachel McClelland, an activist and filmmaker who organised the letter, told Sky News seeing ministers defend Mr Abbott left "me and many other activists with our heads in our hands".
Labour leader Sir Keir Starmer said he also had "real concerns" about Mr Abbott and "I don't think he's the right person for the job".
Lesbian, gay, bisexual and transsexual rights charity Stonewall also expressed concern about the potential appointment of Tony Abbott as a United Kingdom trade envoy.
Mr Hancock replied: 'I don't think that's true'.
Abbott has also said he feels "a bit threatened" by the LGBTQ+ community during an interview with Chanel 9's 60 Minutes, adding "as so many people do ... it's a fact of life".
Tory MP Caroline Nokes criticised Mr Abbot's track record on women and rubbished a reported plan by the British government to appoint him to a new trade board.
Shadow global development secretary Emily Thornberry said Mr Abbott was "the wrong" choice "on every level" and had "no experience of detailed trade negotiations, no understanding of Brexit, no belief in climate change, no concern for workers' rights".
"By his own admission, his contribution to Australian free-trade agreements was to tell negotiators not to sweat the technical details and to just get them done", Grozoubinski tweeted.
"If the government is truly committed to an outward-looking future for Britain, to tackling climate change and to creating an equal society for all, it should reconsider its proposed appointment", the letter said.
"So does this indicate the government is moving away from any commitment in trade deals to maintain environmental protection?"
Tony is tipped to be the UK's new trade envoy, but the decision has been slammed by politicians who have labelled him a misogynistic, homophobic, climate change denier.
"We needed experts in different areas and somebody who is the former Prime Minister of Australia is obviously an enormous expert in the field of trade".
During that time, and his tenure as leader of the opposition between 2009 and 2013, Abbott received flak over controversial statements and stances, such has his opposition to same-sex marriage, and a remark that it's not necessarily "a bad thing" that women are in fewer positions of political power because men may be "more adapted to exercise authority or to issue commands".
Scotland's First Minister Nicola Sturgeon has joined the voices of criticism, saying Mr Abbott was the wrong man for the job.
Number 10 has so far declined to comment on the "political debate" surrounding Mr Abbott while ministers have insisted no appointment has yet been made.