It brings to an end months of speculation over Mr Rudd's bid for the role, currently held by Ban Ki-Moon.
Ms Bishop said it would be a matter for the Cabinet.
"Kevin Rudd has requested that the Australian Government nominate him and, as the Prime Minister has indicated on a number of occasions, that'll be a matter for Cabinet," she told Sky News.
"I'll certainly put the matter forward."
When asked if he supported the nomination of Mr Rudd, Prime Minister Malcolm Turnbull replied: "As the Foreign Minister has said, and I think I've said on many occasions, this is a matter that we will consider in the Cabinet."
Opposition Leader Bill Shorten urged the Coalition to back Mr Rudd, saying he was "suitably qualified".
Mr Shorten added that it would be "petty" for them to not support Mr Rudd's bid.
"I think Mr Turnbull should put the national interest first and support an Australian," he said.
"The Turnbull Government has to decide if it's an Australian Government or a Liberal Government.
"If it's an Australian Government, it will support the Australian. If it's a Liberal Government, it won't back the Australian because of petty political reasons.
"Australians have had enough of that."
Previously, South Australian Liberal senator Cory Bernardi had urged Ms Bishop not to endorse a "dysfunctional", "vengeful", "unstable", "megalomaniac" like Mr Rudd.
Senator Bernardi told the ABC in February that "the Australian public would be very disappointed if we endorsed such a person for this significant role".
If nominated, Mr Rudd will face former New Zealand prime minister Helen Clark, who announced her candidacy earlier this year.