Both clubs have previously been reluctant to go down the route of sharing a stadium, however Elstone has revealed Everton are not against the idea, if it makes financial sense.
He told Sky Sports News: "It's certainly one of the options that we will need to cover.
"A shared stadium is perhaps an option if it's affordable.
"We have to look at where we can raise money, because potentially Liverpool will have to obviously contribute to that, and Liverpool City Council perhaps might need to find some money.
"Our history is one of creativity and innovation and if we are the first major English club to look at sharing then we're not scared of making those decisions."
Liverpool's plans to build a new £350million stadium in Stanley Park have been hit be a number of delays and the project was put on hold due to the worldwide recession last year.
Liverpool's deputy executive director Peter Shaw played down the possibility of the city's two clubs sharing a stadium.
He said: "It's not on our agenda at the moment.
"Liverpool are progressing forward with our own stadium. That is the position we are still in.
"The LFC stadium is quite far progressed and once the financial markets reopen for business the LFC stadium will progress further."
When asked whether the idea of a groundshare with Everton could be a possibility, he said: "That's not for me to answer."
Everton's plans for a new stadium at Kirkby were rejected by Communities and Local Government Secretary John Denham.
The club and Tesco had hoped to build a new stadium and a shopping complex, in a move which would have seen Everton leave their long-time home, Goodison Park.
The Government announced in August 2008 that the stadium plans would be 'called in'.
The planned development, including a 50,000-seat stadium, was then examined during a public inquiry which took place between November 2008 and February 2009.
A majority of Everton supporters - 59.27% - had backed the move and Knowsley Council were also firmly behind the project.
While sharing with Liverpool would be a controversial step, it may also be the most sensible.
There had been opposition to the Kirkby proposal on the grounds it would move the club four miles away from Goodison, and outside the city boundary.
In the decision note, explaining the rejection of the proposal, the Secretary of State agreed with the Inspector that the impact of a football stadium on the chosen site "would have a harmful effect on the living conditions" of some nearby residents due to "loss of daylight and harmful visual impact", and to other locals on the basis of "noise".
It was also stated that the departure from the stadium of supporters after matches would "have a significant effect on the way in which the town's residents conduct their lives during that time".
The decision can be challenged by a letter to the High Court within the next six weeks.
There could still be hope for Everton if they remain keen on Kirkby, with Denham disagreeing with the Inspector's view that Kirkby cannot house a football stadium.
In the decision note, it is stated that "the Secretary of State does not agree with the Inspector's conclusion that this suburban residential town is not a suitable location for such a large football stadium".
The note continues: "This is because, whilst he shares the Inspector's concerns regarding living conditions and those other concerns identified regarding the stadium, the Secretary of State does not consider that this necessarily precludes an alternative proposal for a stadium within the town of Kirkby coming forward which might be acceptable."
Elstone said: "If the scheme was rejected in a way that we can amend it or tailor it to give it greater chance of success then that's something clearly we'll look at.
"I'll sit down with the partners and review the decision before we make our next steps on that particular point."
Elstone ruled out renovating Goodison instead of finding a new ground, saying: "It's a very small site, it's locked in by houses and business and a church and a pub and a school, so to redevelop would be incredibly challenging and when something is challenging and perhaps ambitious the other word that often goes with that is 'expensive'.
"We believe the redevelopment of Goodison would be very expensive. Future stadium potential for Everton is probably not necessarily about land, it's more about cash, the money, the affordability and that was the beauty of the Kirkby scheme which did come with a substantial subsidy from Tesco.
"Any alternative is going to have to address affordability. I would say as well that Kirkby for a long, long time has divided the club and divided the fans, and who knows really the proportion of fans that were for or against it? We don't know that."
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