Bhandari, 70, and Greenwood are seeking re-election at the Hague-based International Court of Justice.
One-third of the court's 15-member bench 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 Thursday, four of the six candidates in the fray were elected as per the UN laws, got absolute majority in both the General Assembly and the Security Council.
Ronny Abraham of France, Abdulqawi Ahmed Yusuf of Somalia, Antonio Augusto Cancado Trindade of Brazil and Nawaf Salam of Lebanon were elected after four rounds of elections on Thursday.
Yesterday, the UN General Assembly and the Security Council met separately to elect the remaining one candidate for the ICJ.
In each of the five rounds of elections, Greenwood of Britain received nine votes and Bhandari got five in the UN Security Council. The winner must secure 8 votes in the Security Council.
Given that Britain is a Permanent member of the Security Council, Greenwood has an advantage over Bhandari.
Bhandari received absolute majority in the General Assembly elections in all the five rounds. In fact, he increased his vote tally from Thursday's 115 to 121 votes in the General Assembly elections yesterday against the absolute majority number of 97.
The vote tally of Greenwood dropped from 76 to 68 on Monday.
Both the General Assembly and Security Council announced to adjourn the meeting for the election to be convened at a later date.
Ahead of the yesterday's voting, Congress leader and former top UN official Shashi Tharoor said that the "voice of the General Assembly" has been ignored for too long.
He accused the "UK of trying to stall the will of the majority of the UN General Assembly".
"As the UN Security Council (SC) & General Assembly (GA) vote to choose a judge for the International Court Of Justice (#ICJ) between Indian & UK candidates, the legitimacy & effectiveness of the UN are at stake. The voice of the GA has been ignored too long," he said in a tweet.
"This time a nominee of a Permanent member of the SC has failed to get an absolute majority of the GA, for the first time in a direct contest to a major @UN organ. GA vote has turned into a protest against an unwarranted extension of privilege for 70+ years. P5 lost by 40votes!" he said.
Tharoor said the election is no longer about the judge or the country he hails from, but about the General Assembly standing up against a member of a privileged club who has lost comprehensively among the Members at large but still leads 9-6 in the Security Council.
"The UK is trying to stall the will of the majority of the GA," he charged.
