DUP leader Peter Robinson and Martin McGuinness of Sinn Fein are expected to consider the allocation of ministries.
The DUP and Sinn Fein won the most seats with 38 and 29 respectively.
Mr Robinson dedicated his party's victory to Ronan Kerr, the PSNI constable who was murdered by dissident republican terrorists last month.
The spotlight since the election has been on the Ulster Unionists, who are one MLA short of being able to claim two ministries in the power-sharing cabinet.
It is understood they are trying to tempt a successful independent candidate back into the fold.
David McClarty, who left the UUP following a row with his local constituency party, will not be drawn on whether he will return.
Responding to UUP leader Tom Elliott's description of Sinn Fein as 'scum' at an election count in Omagh, Mr McClarty said that it was not indicative of the kind of "progressive unionism" he believed in.
However, Mr McClarty said he had not discussed that issue with Mr Elliott during "amicable talks" on Sunday evening.
In the last assembly, Alliance held the justice portfolio following a cross-community vote procedure introduced in the Hillsborough Agreement.
If Mr McClarty is not attracted back into the UUP fold, the figures suggest Alliance could be entitled to a ministry by right.
It raises the prospect of the UUP on 16 assembly seats having one executive ministry while the Alliance, with eight seats, has two.
The discussions on Monday will take place as counting begins in the council elections in Northern Ireland.
Almost 600 positions are to be filled across 26 councils, in the first poll since 2005.
Mr McGuinness said he hoped to meet Mr Robinson on Monday, but a DUP source said the talks would depend on how swiftly the Castlereagh count was completed.
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