Officials say hosting the event would cost the city council about 95,000 – around one-third of the total 285,000 bill – but estimate it could boost the Capital's economy by 1 million.
The sport's world governing body, the International Archery Federation, has extended the closing date for bids specifically to allow a bid from the Capital.
Councillors are due to consider the proposal to bring the event, in 2010 or 2011, on Thursday.
Previous target archery World Cup Finals have been staged in iconic venues like the Mayan pyramids in Mexico and Burj-al-Arab in Dubai.
Run over seven days in October or November, it culminates with an Olympic-style award presentation.
If Edinburgh does mount a bid, a local organising committee would be formed involving the city council, EventScotland, UK Sport and the Grand National Archery Society.
A report by Jim Inch, the council's director of corporate services, says EventScotland has already promised funding of up to 90,000. The council's contribution would be spread over three years. The host city has the right to retain all income from merchandising, sponsorship, sales and media.
The report points out there is no funding in the council's budget for the event at present, but if Edinburgh made a successful bid the costs might be covered by "capital city status funding".
Deputy council leader Steve Cardownie, who is in charge of events for the city, said he had yet to discuss the archery proposal with officials.
It is not clear exactly where the event would be held but Cllr Cardownie said Peffermill playing fields and Meadowbank sports stadium were possibilities.
But he said the decision on whether to make a bid would be based on whether hosting the event would be value for money.
He said: "We are quite mercenary about it – what does it do for our economy, does it boost our international profile and can we afford it?
"We are constantly on the look-out for events that boost the city. But bearing in mind money is tight, we have to be convinced these events are winners and I don't know if this one hits the bull's eye."
However, independent Lothians MSP Margo MacDonald, former PE teacher and convener of the Scottish Parliament's cross-party group on sport, said hosting the final was a good idea.
She said: "If it was held somewhere accessible I think people would go along to see it. And if there was the chance for people to try it themselves that would be an added attraction. I think it could be quite successful."
William Marr, a field archer from Gilmerton who was taking part in yesterday's Scottish Championships in Musselburgh, said he would welcome the event coming to Edinburgh.
He is due to begin target archery this year, and he could feature in any upcoming Edinburgh championships.
He said: "Any event like this coming to the city is great.
"I'm not sure where it would be held, when we do heats for outdoor events it has to be at Penicuik high school because it doesn't have any roads behind it. For the longer heats you need 90m so football grounds are too small, plus there are also spectators to think about as well."