These threats were made after her criticism of two resolutions adopted last week at committee stage ahead of a final vote, Carmel Shama Cohen asserted on Israeli public radio.
The resolutions refer to "Occupied Palestine" and the need to "safeguard the Palestinian cultural heritage and the distinctive character of east Jerusalem".
Israel suspended cooperation with UNESCO on Friday, a day after its committee adopted the two resolutions.
The two resolutions are due to be put to the executive board of the United Nations Educational, Scientific and Cultural Organization on Tuesday, which generally votes with the line taken by committees.
Israel occupied Jerusalem during the 1967 Middle East war and later annexed it in a move not recognized by the international community.
Source: World Bulletin