Woolcott urged the Australian government to formally nominate Rudd for the position, following Foreign Minister Julie Bishop's confirmation that Rudd had approached the government for endorsement, Xinhua news agency reported.
Woolcott served as Australian Ambassador to the UN from 1982 to 1988, and President of the UN security council for Australias term in November 1985, The Guardian reported.
Woolcott told the media that the government should support an Australian endeavour, regardless of which side of politics he sat.
"If an Australian decides to stand I think the Australian government should support that Australian," Woolcott said.
Rudd has a strong relationship with China, while he would also fare well when dealing with other, western nations, Woolcott added.
Woolcott said: Rudd is well-respected and well-known by Xi Jinping, the Chinese Head of State. I think the same would probably apply to Britain, France and the US.
His problem could be Russia because of the way the relationship with Russia deteriorated when Tony Abbott was Prime Minister when he was talking about shirt-fronting Vladimir Putin, The Guardian quoted Woolcott as saying.
Thirteen candidates have been formally endorsed, including former New Zealand, Prime Minister Helen Clark, whom many government MPs would prefer to see as UN Secretary-General in place of Rudd.
The government is yet to decide if Rudd would be nominated but a decision is expected either on this or next week.
Woolcott dismissed concerns about Rudds suitability, including from former Labour premier Kristina Keneally who said many in parliament would oppose Rudd and her labrador would be more suitable, The Guardian reported.
Incumbent UN Secretary-General Ban Ki-moon will step down from the position on December 31.
This website uses cookies.