Merlin operates 99 attractions in 22 countries, and has 54 million visitors to its parks a year.
It says it is the second-largest company of its kind in the world after Walt Disney.
Its other well-known attractions include Alton Towers, Thorpe Park and Warwick Castle.
The business has a turnover of Â£1bn a year and is thought to be worth some Â£3bn.
The share offer, which will see 20% of the company pass to public investors, is accessible to smaller shareholders with a minimum investment stake of Â£1,000. Shareholders will also get a one-off 30% discount on certain entry passes for Merlin's theme parks.
The company, which was formed in 1999, says the number of visitors to its parks has grown by 11% a year since 2008.
The company's non-executive chairman, Sir John Sunderland, said: "I have been impressed by the significant growth that Merlin has delivered as a private company.
"But there is more to come and I believe Merlin has a very promising future as a publicly listed company."
Merlin Entertainments, the UK company that owns Legoland, Madame Tussauds and Chessington World of Adventures, says it plans to issue shares to the public.