Note: I can’t understand the original Hebrew source in the Zohar but assume the reference is correct.
“Once the rabbi, the Hasid, our rabbi and teacher Menahem Nahum of Chernobyl was in our town, and he told this story to the people. I came in the middle of the story, and I heard his version which contained some changes and something new—that the Messiah said to the Besht, ‘I do not know whether you will open the gate, but if you do redemption will certainly come to Israel.’ He said further that this was the gate of the palace of the Bird’s Nest through which no one has ever passed save the Messiah, as it is said in the holy Zohar.  He said that he heard God’s voice saying to him: ‘What can I do with you since I must fulfill your will?’In Praise of the Ba’al Shem Tov [Shivhei HaBesht], https://archive.org/details/inpraiseofbaalsh0000dobb/page/58/mode/2up?q=nest
Source for Tale 42. The Palace of the Bird’s Nest
“The Zohar, III, 20-26 [II 7b-8], V, 282 [III, 196b]. The idea of the Messiah being in the heavenly palace called the Bird’s Nest is found in the earliest portion of the book of the Zohar, “Midrash ha-Ne’elam,” which was written between 1275-80. See Scholem, Major Trends in Jewish Mysticism, p. 188.”
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Zohar Chadash, Midrash HaNe’elam Al Eichah (the original source where it talks about the Messiah being in a “Heavenly palace called the ‘bird’s nest’”) – Hebrew https://www.sefaria.org/Zohar_Chadash%2C_Midrash_HaNe’elam_Al_Eichah?ven=Sefaria_Community_Translation&vhe=Zohar_Chadash&lang=en
By Dr. Dannielle Blumenthal (Dossy). All opinions are the author’s own. Public domain.