Mr Bainimarama, a former coup leader who was elected last year in Fiji's first elections in eight years, said it was "time to dispense with the colonial symbols".
The Commonwealth Pacific nation gained independence from Britain in 1970.
New Zealand, which also has the symbol on its flag, will hold a referendum on whether to change it in 2016.
The current Fiji flag is light blue with the union jack - also known as the union flag - and a shield with the cross of St George and a lion. It also features a sugar cane, bananas, a palm tree and a dove of peace.
"We need to replace the symbols on our existing flag that are out of date and no longer relevant, including some anchored to our colonial past," Mr Bainimarama said.
"The new flag should reflect Fiji's position in the world today as a modern and truly independent nation state."
After a national competition, the new design will be unveiled on 10 October, the 45th anniversary of independence, Mr Bainimarama said.
He pointed to the fact that only four sovereign states which were once British colonies still retain the union jack as a national symbol: Australia, New Zealand, Tuvalu and Fiji.
New Zealand said last October that it would hold a referendum this year to decide on an alternative flag, and another referendum in 2016 on whether to officially adopt the new design.
After New Zealand announced its referendums, Australia's Foreign Minister Julie Bishop said there was "no great demand" in Australia to change the national flag.
Fiji is to remove the UK's union jack from its flag, Prime Minister Voreqe Bainimarama has announced.