When questioned on Sky News, the Health Secretary said he does not "think it's true" that Mr Abbott is homophobic or misogynistic, and added: "He's also an expert in trade."
Mr Abbott is in talks to become joint president of the relaunched Board of Trade but critics have raised numerous concerns over opinions he has voiced - including his belief that coronavirus restrictions should be lifted.
Mr Hancock, wearing an NHS badge with the LGBTQ rainbow flag, was pressed over the suitability of Mr Abbott.
The former Australian PM led his nation from 2013 until he was ousted by his own party in 2015.
Asked if Mr Abbott is a "fit and proper person to represent us on trade" Mr Hancock avoided the question, citing his role as Health Secretary but added: "I do know that Mr Abbott is very good and very experienced at trade".
He added: "It is clearly a very important decision that the Prime Minister and the Trade Secretary are across".
Equalities Minister Liz Truss, when quizzed by ITV News on the matter, offered similar answers to Mr Hancock. She said: "What I’d say about Tony Abbott is he’s a former PM of Australia".
Ms Truss refused to answer whether she was "comfortable" working with Mr Abbott.
Opposition MPs were highly critical of Mr Hancock's defence.
Labour leader Sir Keir Starmer told ITV News he has "real concerns" about the potential appointment of Mr Abbott to the role.
Sir Keir added: "I don't think he's the right person for the job and if I was Prime Minister I wouldn't appoint him."
Shadow minister Peter Kyle saying ministers "don’t even pretend not to embrace bigots anymore".
Labour MP Marie Rimmer added: "Surely there’s trade experts who aren’t homophobic and misogynists? Britain deserves better than Tony Abbott representing us on the world stage."