The Great Exhibition of the North will get Â£5m, with "public and private funders" providing any further budget.
The Department for Culture, Media and Sport said a Â£15m "legacy fund" to attract cultural investment in the North would also be provided.
The competition will run until June, with the winner announced in October.
A board has been set up to handle applications to host the two-month show, chaired by the chief executive of Welcome to Yorkshire, Sir Gary Verity.
Sir Gary said he hoped to see bids from the "many towns and cities across the north of England that have what it takes to host the Great Exhibition of the North".
"We will be showcasing our art, design and culture to the world in 2018 and high quality bids are an essential part of ensuring that we deliver an incredible Great Exhibition."
The funding for the exhibition was confirmed in March's Budget, having originally been proposed in 2014's Autumn Statement.
A competition for towns and cities in northern England to host a Government-funded exhibition promoting northern art, culture and design has opened.