Mr Bhandari and Greenwood, who are seeking re-election at the Hague-based ICJ, are locked in a major battle as 11 rounds of elections held in both the UNGA and UNSC have failed to yield results.
Under the election procedures, the balloting would be held simultaneously by the Assembly and the Council. In the successive rounds of elections spread over two days in the last two weeks, Mr Bhandari, 70, enjoyed nearly two-third majority in the 193-member Assembly.
Mr Greenwood, 62, received nine votes as against five by Dalveer Bhandari in the Security Council. As per ICJ rules, the candidates need to gain majority in both the General assembly and Security Council to be declared elected.
On Monday, the election would be presided over by the General Assembly President Miroslav Lajcak, his spokesperson Brendan Varma said at a news conference at the UN headquarters in New York.
The spokesperson was asked whether the election process would be turned over to the ICJ in The Hague if the General Assembly and Security Council could not agree on the last seat.
He said that there were additional procedures that could be followed in New York if the Monday's meeting remained inconclusive.
For example, a joint conference could be formed, consisting of six members (three appointed by the General Assembly and three by the Security Council). "This joint conference could then agree upon a name by an absolute majority and submit it for the acceptance of the Assembly and the Council," said Mr Varma.
"Nevertheless, the time for that procedure had not come yet," he said and advised the journalist to wait and see what happened in the Monday's meeting.
The ICJ has a bench of 15 judges, five of whom are elected every three years for a nine-year term, elections for which are held separately but simultaneously in the United Nations General Assembly and Security Council in New York.
Last week, four of the six candidates in the fray were elected as they got absolute majority in both the General Assembly and the Security Council.
They are Ronny Abraham of France, Antnio Augusto Canado Trindade of Brazil and Abdulqawi Ahmed Yusuf of Somalia and Nawaf Salam of Lebanon.