BBC Scotland claims it has seen documents drawn up by a consultancy firm hired by the SFA proposing the move.
If given the go ahead, it will signal the end of international matches and cup finals at the national stadium.
Hampden, the home of Scottish football for the last 111 years, is owned by League Two outfit Queen's Park.
However, the SFA holds a lease on the 52,000-seater arena which will expire in 2020.
The stadium was last redeveloped in 1999 at a cost of £59million and the SFA are now examining whether to commit themselves to another rental agreement with the Spiders, or to even acquire the free-holding of the ground.
A source close to the governing body has said that "every option is being considered" - including full or partial redevelopment of the existing stadium.
But plans to quit the ground on Glasgow's south side are also being looked at.
However, a source insists Scottish hopes of landing three group games and a round-of-16 tie for the 24-team Euro 2020 tournament - which will be staged across Europe - are not under threat.
"It is important to stress that UEFA still sees us as a safe bet for Euro 2020," they said.
The BBC claims Celtic Park, Pittodrie Stadium, Ibrox Stadium, Easter Road and Murrayfield Stadium are all under consideration as venues for Scotland's home matches and Scottish Cup games after viewing documents drawn up by James Watson Consultancy.
Hampden also houses offices for both the SFA and the Scottish Professional Football League, as well as the Scottish Football Museum.
The stadium was not used for either of this season's two domestic cup finals after it was handed over to the organisers of Glasgow 2014, who have converted it into an athletics arena ahead of this year's Commonwealth Games.
SFA chief executive Stewart Regan, currently attending the FIFA Congress in Brazil, responded to the reports by tweeting: "A lot of speculation around the future of Hampden Park.
"At this stage the Board is simply doing it's job, considering the implications of renewing a lease that expires in 2020. No decisions made or options considered. High level consultancy only."