The pair have discussed the impact of gender on their political careers and plan to collaborate in changing the perceptions of female leaders as unlikeable, selfish and ruthless.
The first female PM of Australia talks about sexism in politics, Hillary Clinton, and the importance of education for girls. (Video by Global Citizen)
"I'm hopeful there are some things we can do together in the future on these questions of leadership and gender, bringing to that possibility some of our shared experiences," Ms Gillard said in an exclusive interview.
"Personally, I think there's a need to deepen the evidence base about women in leadership," she added, saying there was already much research on the role of 'unconscious bias' in attitudes to female political leaders.
"I think there's a need for more of that research and for it to be brought together to help people understand it and then action change, and I think that is a journey that lies before all of us."
Ms Gillard agreed to be interviewed about being Australia's first woman prime minister as she prepares to join John Howard, another former prime minister, and others in judging what may well be Australia's first national award for political leadership, the McKinnon prize.
Just as she said that gender explained some things, but not everything, about her prime ministership, Ms Gillard said the same went for Ms Clinton and her defeat by Donald Trump in the 2016 US presidential election.
"I think there are many lessons coming out of the Trump/Clinton context and, because the whole campaign was not about gender, the lessons are about more than gender," Ms Gillard said.
"But I think there would be quite a few people in the States in various walks of life who would be reflecting on how they formed their impressions of the candidates and the quality of the information they had and the judgments they made, and probably concluding to themselves, 'I'm not going to easily accept some things again that I accepted last time.'
"I think that is going to stand democratic decision making in good stead for the future, and I think there is a gender element to that."
Both Ms Gillard and Ms Clinton devote a chapter to the issue of gender in politics in their memoirs. Both refer to the research of Facebook chief operating officer Sheryl Sandberg, who has shown that the more successful a man is, the more he is liked, and how it is the opposite for women.
"Sheryl shared another insight: that women are seen favourably when they advocate for others, but unfavourably when they advocate for themselves," wrote Ms Clinton.
Describing sexism and misogyny as endemic in America, Ms Clinton writes that misogyny is "what happens when a woman gets a job that a man wanted, and instead of shaking her hand and wishing her well, he calls her a bitch and vows to do everything he can to make sure she fails".
Ms Gillard said a starting point for changing perceptions was for people to ask themselves why they think the way they do.
"One thing that is very commonly believed is that once one woman gets through, that that woman tends not to help the women behind her come up through to the same level she is.
"Sheryl [Sandberg's] research debunks that, and yet I talk to a lot of people in business and it is very commonly believed. It's very commonly believed by women who are aspiring to get to the top of business."
Ms Gillard believes this perception stems from an "unconscious bias that women who get to the top aren't very likeable".
"Let's be knowing about that and thoughtful, and if we are we'll get rid of the stereotyping and the sexism that comes with it," she said.