It is understood Peter Robinson and Martin McGuinness met on Monday afternoon to discuss the next steps towards the formation of an executive.
Sources told the BBC that the DUP and Sinn Fein are keen to try to fill ministerial positions on Thursday.
However, it is believed Mr Robinson wants to have a round-table meeting with all the Stormont party leaders.
The DUP and Sinn Fein won the most seats in Thursday's NI Assembly election 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 in Omagh 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 Alliance, with eight seats, could have two.