The proposal would see the carrier docked next to aquarium The Deep, for Hull's City of Culture celebrations in 2017.
The 20,000-tonne carrier is due to be retired, after 32 years of service.
Mr Johnson, the Hull West Labour MP, said there was a "pressing need" to secure the future of the vessel.
As part of the plans, the 700ft long (215m) ship would be based on the River Tyne, at the former Holburn Dock, for the "next two years" before berthing in Hull for two years, from 2017.
A statement on the MP's website said if the plans were a success, HMS Illustrious would be available as an "immersive, inspiring and exciting visitor experience" during 2017 and 2018.
A Ministry of Defence spokesperson said: "HMS Illustrious has a long and proud history with the Royal Navy.
"During her 32 years of service she has protected our nation's interests in the Falklands, Bosnia, Iraq, Sierra Leone and most recently the Philippines."
The ministry was considering bids to re-home HMS Illustrious and preserve the carrier as a "lasting tribute" to the personnel who served on all three of the Invincible-class aircraft carriers, introduced in the 1980s, the spokesperson added.
It is a condition of sale that HMS Illustrious must remain in the UK, the ministry said.
Last month, the ship returned to its home port in Portsmouth for the final time.
Plans to turn the Royal Navy aircraft carrier HMS Illustrious into a tourist attraction in Hull have been backed by MP Alan Johnson.