The club has launched a consultation on quitting Gorgie for a new 45 million state-of-the-art facility with a 30,000 capacity. If given the nod, bosses hope to submit a planning application to build a new stadium within the next 12 months, with a view to move into the facility by 2014.
Hearts chiefs are considering Sir David Murray's proposal to build a stadium on green belt land in west Edinburgh, but said they were working with a "blank piece of paper" and were open to all suggestions.
A senior Hearts source said: "The club met with all key supporters groups last night to introduce the plan and it was broadly accepted that the club does need to look at alternative sites to Tynecastle."
The cost of building a new stand at Tynecastle is estimated at 20m while a new stadium elsewhere is expected to cost between 30m and 45m.
Hearts fan and deputy city council leader Steve Cardownie said: "Things in the development world have moved on, so I'm not surprised if Hearts have come to the view the best solution is to move to a new stadium.
"Tynecastle is a stadium built for the last century - no car park, surrounded by tenements. It will be a shame if we have to go, but I know they genuinely wanted to stay at Tynecastle, so if they deem it necessary to move, it will be necessary."
Many comments from fans on internet forum Jambos Kickback also seemed open to the idea of a move.
One said: "If moving means that we can prosper somewhere that isn't Tynecastle then so be it. At the end of the day Tynecastle is Hearts but Hearts isn't Tynecastle and I'll follow Hearts anywhere."
Another said: "In a perfect world we would stay at Tynecastle but it's not."
Former Hearts chairman George Foulkes said he was ready to consider and debate all options, but his own preference was to stay at Tynecastle.
He said: "My initial reaction is to support staying where we are, to feel we are part of the community there and the ethos of the club depends on being there, but it depends on where the alternative site is.
"I suspect what is being planned is David Murray wants to breach the green belt and has been trying for some time to get a development on the site he owns on the outskirts of the city. I'm not keen on breaching the green belt and I'm not keen on a peripheral site."
Club sources said fans had to "seriously consider" the team's future, but stressed that the club would not be heading to Murrayfield under any circumstance.
They said, however, that it was in the club's best interests to build a new home as the current stadium's main stand could soon be beyond repair.
It is not yet known whether the stadium will be sold off.
Bosses, however, are keen to get the blessing of Hearts fans.
The source added: "We haven't ruled out talking to Sir David Murray about his green-belt project, but we are definitely not going to Murrayfield.
"Bosses will listen carefully to what the fans have to say. If they said no, we would stay here and invest in maintenance, but we really need to look decades ahead."
A consultation offering fans details on the potential move and a questionnaire were published on the Hearts website this morning. In forthcoming weeks, Hearts bosses will meet with the board, club stakeholders and fans.
A spokesperson said: "We will be seeking the views of our supporters. We fully appreciate the emotional attachment fans have to the stadium, however it is important that we lay the foundation stones for generations of Hearts supporters in future."
NO PROGRESS IN THREE YEARS
HEARTS' plans to redevelop Tynecastle have still not progressed nearly three years after they were first announced.
And last year, the 51 million scheme was scaled back by delaying the building of commercial and leisure facilities as part of the project. A club source said: "We've been here for 125 years and our main stand dates back to 1914, but it has become increasingly difficult to fund repairs to it.
"We'd like to knock down this main stand and replace it with a 10,00 capacity stand and function suites, but the club is facing huge difficulties due to strict health and safety guidelines."
The nearby North British Distillery makes potentially dangerous ethanol.