The Scottish FA will choose between Hampden and Murrayfield as the home of the national team this summer.
A proposal to hold international matches and cup games at Celtic Park or Ibrox was rejected at a specially convened board meeting.
The SFA's 20-year lease on Hampden expires in 2020, with a renewal option.
The governing body commissioned a feasibility study to consider all of its options for moving away from the national stadium.
The study also considered the pros and cons of staying at Hampden - that case being put by its owners Queen's Park FC.
Celtic Park and Ibrox each hosted one match during the Euro 2016 qualifying campaign, and internationals have been held at both grounds previously.
However the SFA board will now focus on renewing their Hampden lease or taking up the offer from Murrayfield, home of Scottish rugby.
"As well as submissions from those venues involved in the process - Queen's Park, Scottish Rugby, Celtic and Rangers - the board also received supplementary information from other key stakeholders, including the Scottish FA membership," said the SFA in a statement.
"Over the course of the meeting, each proposal was discussed in detail and broad themes were developed and debated among the group. The request from Glasgow City Council to review the timescales of any decision was also acknowledged.
"Having considered all submissions and supporting documentation, the board has decided to further develop two positive options: remaining at Hampden Park beyond 2020 and moving to BT Murrayfield. This decision was taken after considering the financial implications of all proposals submitted.
"Given the complexity of the discussion - as well as the social, economic and emotional impact of any final decision - the board was unanimous that more work requires to be undertaken to bring further clarity and certainty to any final decision."
Paul Goodwin, co-founder of the Scottish Football Supporters Association, says the choice must be between the historical significance of Hampden and the needs of football supporters.
"Part of the problem we have with the stadium [is that] we're emotionally attached to it for [being] steeped in the history of Scottish football," Goodwin told BBC Scotland.
"But it doesn't have the quality, it doesn't have the transport, the infrastructure. Everyone is really divided over it, so it's having a range of options on the table and deciding what's for the good of Scottish football."
Peter Dallas, the managing director of Hampden Park, has been encouraged support for Scottish football's traditional home.
"We note the Scottish FA board statement and look forward to working with the board, other key stakeholders and the working group, once it is established, as we develop a final proposal in line with quoted timescales," Dallas said.
"It has been heartening to receive so much support during this process to date, from supporters, stakeholders, sponsors, the media, local council and MSPs.
"That underlines the special place Hampden Park holds in the hearts and minds of so many as the spiritual home of Scottish football and we are looking forward to reaffirming its position as Scotland's National Stadium beyond the matches being hosted here for UEFA Euro 2020."
A spokesperson for Celtic said: "We were asked to consider a 20-year commitment and Celtic was willing to enter a transparent, open-book partnership process, to assess the suitability of Celtic Park to host matches.
"Clearly, the SFA has rejected this approach and chosen not to consider Celtic Park as a potential venue. This is something we accept and we wish the association well in pursuing their other options."
Former Scotland international Michael Stewart believes one solution may be to rebuild Hampden, reducing the capacity to 30,000-35,000, for crowds that only partially fill the country's largest stadia.
"How about trying to redevelop Hampden and make the capacity lower and use Murrayfield and Hampden?" Stewart told BBC Sportsound.
"How often do we have a half-empty stadium? We [could] close off the ends at Hampden and make it a top-drawer 35,000-seat stadium and use Murrayfield for the bigger games.
"We don't want to be hostages to the history of Hampden, we need to look forward to what is best for the game. I don't think Hampden is fit for purpose but what is the alternative?
"Behind the goals at Murrayfield, the area is too big, the main stand is far from the pitch, but there is a lot more going for Murrayfield than Hampden, with the transport links.
[With] all the facilities at Murrayfield, there's greater scope for higher revenue. If the SFA were using it, they have a better opportunity to generate more money than at Hampden."