It comes after the Northern Ireland deputy first minister Martin McGuinness resigned, effectively ending power sharing at Stormont on Monday.
The majority of people in Northern Ireland, 56%, voted Remain in the European Union referendum, a stance backed by McGuinness' Sinn Fein party, although the British province is ruled by the Brexit-supporting Democratic Unionist Party, (DUP) which Foster leads.
The government, formed under terms of Northern Ireland's 1998 Good Friday peace accord, requires support from McGuinness' Irish nationalist Sinn Fein party and Foster's Democratic Unionists, who represent the British Protestant majority.
Mrs Foster has announced plans for a public inquiry into the RHI affair while her party colleague, Economy Minister Simon Hamilton, has sent proposed measures to slash the £490 million bill to Sinn Fein Finance Minister Mairtin O Muilleor for consideration. Sinn Féin will not tolerate the arrogance of Arlene Foster and the DUP. Many fear that Northern Ireland will have no government in place at arguably the most important moment in its modern history: Brexit. So with McGuinness's resignation, Foster must also now step down.
Northern Ireland is facing its worst political crisis in at least a decade, as a senior minister's shock resignation looks set to dissolve the government, threatening the Good Friday power-sharing agreement that has helped keep peace for nearly two decades.
Under this scheme, launched by the Northern Ireland Department for Enterprise, Trade and Investment in 2012 and equivalent to a scheme in Great Britain, businesses and other non-domestic users were offered a financial incentive to install renewable heat systems on their premises.
McGuinness explained in his resignation letter that his party, Sinn Féin, would not be nominating another candidate to fill his role.
Mrs Foster said: "I am disappointed that Martin McGuinness has chosen to take the position he has today".
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"So it stands to reason that if there is no Northern Irish assembly, and there is no Northern Irish executive for much of the time before the March timetable she has set before invoking Article 50, then she will be unable to properly consult, to fully discuss and to find agreement on the complex issues during this time period".
The British government has said that it will try to facilitate an agreement between the DUP and Sinn Fein in the coming days but few believe such a compromise is possible.
The RHI meant to subsidise eco-friendly boilers, when Foster was economy minister.
An election now is probably the last thing anyone wants to see in Northern Ireland.
The issue that precipitated the collapse is the so-called "Cash for Ash" scandal whereby a scheme to incentivise the use of green energy was mismanaged, with major financial implications for the public purse (to the tune of £490m).
In a tweet, Sinn Fein president Gerry Adams praised Mr McGuinness for his "valiant service".
Speaking after his meeting with Mr Kenny, Deputy Adams said: "As co-equal guarantors of the Agreement the Irish and British Governments have a responsibility to defend and implement the Good Friday Agreement".