The tournament, called the Bingham Cup, has been contested since 2002, when it was created to raise awareness about homophobia in sport.
The Sydney Convicts, one of the most successful rugby union teams in the world, are supporting the bid, which will be submitted this week with the support of Gillard.
"Australia is a sports-loving nation which also has an abiding commitment to diversity and mutual respect," said Gillard.
"In recent years, Sydney has hosted the gay games, the Olympics and the Rugby World Cup with a remarkable spirit of openness and celebration.
"I know that Bingham Cup participants would receive the same warm and generous welcome."
Convicts founder and bid committee chairman Andrew Purchas will look for further support from Australia's senior national side, The Wallabies, as well from state government officials.
"It is very clear that homophobia is quite rampant in the footy codes and many of the Convicts players would not be playing rugby if it was not for the Convicts," said Purchas.
The 2012 edition of the tournament was held in Manchester earlier this year, when the Convicts claimed the title. It has previously been hosted in San Francisco, London, New York, Dublin and Minneapolis.
The Manchester event's website claims the Bingham Cup is largest men’s 15-aside rugby union tournament in the sporting calendar outside of the IRB Rugby World Cup.
It attracts over 30 clubs from 15 countries with approximately 1,500 players taking part.