Crowds of up to 16,520 are expected to be able to watch each session of the 6-day championships involving around 1,000 gymnasts. Britain last staged the championships in Birmingham in 1995.
The 2009 event would provide the sport with a much-needed boost in the UK after its Lottery funding was cut in February because of poor performances at last year's Olympics.
It would also help the Olympic bid since London has not staged as many high-profile championships as Paris.
Details of the Dome's transformation from a notorious white elephant into an entertainment venue were revealed this week. The U.S. company, Anschutz Entertainment, is heading a £500million investment in the site.