Spurs confirmed this morning they had officially registered an interest in moving to the ground after the 2012 Games just hours before yesterday's deadline and hours before being given the green light to build a new stadium of their own.
Levy admitted the club were covering their bases - a move which appears to have been prompted by fears a wrangle with Transport for London will see their plans to construct a £400million ground vetoed by London Mayor Boris Johnson.
TfL, chaired by Johnson, were one of the key objectors to Tottenham's planning application.
They are demanding Spurs contribute £3million to upgrade the Tottenham Hale railway interchange but the club are refusing to pay more than £2million.
That did not stop Haringey Council's planning committee unanimously voting to approve the scheme following a four-hour meeting last night.
Spurs boss Redknapp believes the planned 56,250-seat ground would be a perfect home for the club, but reckons the Olympic Stadium in Stratford - which is likely to have 60,000 seats for football purposes - would also be ideal.
He joked: "I don't want to say the wrong one and Daniel goes, 'What's he talking about?'
"New stadium would be great, or the Olympic Stadium would be great.
"It's not up to me; I'm only an employee."
He said of the new stadium, dubbed the Northumberland Park Development Project: "I spoke to the chairman this morning and, if we've got permission, that's fantastic for the club."
Of the prospect of relocating to the Olympic Stadium in Stratford, he added: "Stratford's fine. There used to be a great pie-and-mash shop there.
"It's up to people who run the club as to where they want to go in the future. It's a long way off still."
Bidding to take over the Olympic Stadium puts Tottenham in direct conflict with West Ham, who also formally registered their interest in doing so yesterday.
Former Hammers player and manager Redknapp said: "It's a difficult one: do West Ham fans really want to leave Upton Park? Probably not.
"They love it there. It's a great atmosphere, fantastic. It's a great little stadium with special tradition."
Levy said in a statement issued this morning: "It is only prudent and good management that we ensure that we investigate all possible options for the club.
"We were informed by the Olympic Park Legacy Company that were we not to register an interest at this time, there would not be an opportunity at any future date.
"We have always maintained that we wouldn't undertake any project that could undermine the overall financial stability and future success of the club and this shall remain our guiding principle going forward and in determining our best option in the interests of the club and all its fans and stakeholders."
He added of the new stadium project: "We shall continue to work with Haringey Council and the application will now be considered by the Mayor of London and referred to the Secretary of State."
West Ham delivered their bid to run the Olympic Stadium after the 2012 Games to 10 Downing Street yesterday, insisting only they could deliver the legacy promise made when London was named host city.