"We think it will be a more successful candidacy," Prime Minister Boyko Borisov said ahead of a government meeting.
Bokova failed to make a strong showing in an informal vote by the 15-member council this week, placing only sixth in the fifth straw poll.
Georgieva is seen as potentially having better chances to succeed South Korea's Ban Ki-moon when he steps down on December 31 as the world's top diplomat.
"Bokova has deployed a lot of efforts. We gave her a final chance for the latest vote... She fell even further and was overtaken by several other countries and other female candidates," said Borisov.
On Tuesday evening, Bokova had still insisted she would not quit her candidacy.
"If (Bulgaria) presented another candidate at this stage, it would be a serious political error," she told Bulgarian media.
Portugal's former prime minister Antonio Guterres is currently in the lead to win the race.
None of the other eight candidates picked up the minimum nine votes required to become the Security Council's nominee for the top post in Monday's straw poll.
The next informal ballot is scheduled for October 5 but the five veto-wielding powers will be using coloured ballots in that round to indicate whether they would block a candidacy.
Council members are hoping to agree on a nominee in the coming weeks and present that candidate to the General Assembly for approval.
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