England's poor World Cup campaign has sparked an inquest into the state of the national team and Hodgson has come in for some fierce criticism after their early exit in Brazil.
Former Manchester United full-back Neville, who boasts 85 England caps, is Hodgson's right-hand man and Owen is confident the 39-year-old could make the step up to the top job.
"If the Football Association is planning beyond the end of Roy Hodgson's current contract in 2016, they should look no further than the man sitting alongside him as the next England manager. Gary Neville," Owen wrote in The Telegraph.
"Neville can spend the next two years being readied for the senior role and see the job through with the current crop.
"If we really want to be bold we have to consider what Germany did when they entrusted Jurgen Klinsmann with the 2006 World Cup campaign, or indeed what Barcelona did with Pep Guardiola.
"There are moments when you should trust your gut instinct that a former player has the knowledge, the respect, the aura and the natural leadership skills to do the job. Neville has these qualities.
"I would be confident of progression – and genuinely optimistic about the direction we will take – if Neville was leading rather than assisting England in Russia in 2018."
Owen also suggested that it would be a mistake for his former Liverpool team-mate Steven Gerrard to retire.
The midfielder insisted at the weekend that he would not rush into a decision over his international future amid reports that he would call it a day for England.
"I do not see why Gerrard cannot leave the door open to be used as and when his country calls upon him," Owen added.
"He does not need to play every minute of every qualifying game, but to discard his experience and desire – especially as he will be invaluable to the likes of [Ross] Barkley over the next few years – would be a waste of his talent."