The website of the United States Department of State suggests that this is not possible:
Individuals with Israeli citizenship, regardless of other nationality, including U.S. citizenship, are prohibited from entering Gaza, entering or departing Jordan via the Allenby/King Hussein Bridge, and are generally prohibited from traveling to parts of the West Bank under PA control (Area A), to include Bethlehem and Jericho.
A later section of the same page elaborates a little:
Travel to and from Jordan: Israeli citizens, including dual U.S.-Israeli citizens, are prohibited from using the Allenby/King Hussein Bridge crossing, unless as part of an official delegation or with special permission from the Israeli and Jordanian authorities. They must cross at the Yitzhak Rabin/Wadi Araba crossing in the south, near Eilat or the Jordan River crossing/Sheikh Hussein Bridge in the north, near Beit She’an.
Source: travel.state.gov information page on travel to Israel.
The website of the Israeli Airports Authority (which manages the Allenby crossing) appears to corroborate this:
The Allenby Border Terminal is reserved for Palestinians and tourists only; Israeli citizens are prohibited from crossing the border here. Indeed, the terminal is used mainly by the Palestinian population from the West Bank, and throughout the years it has operated in tight cooperation between Israel, the Palestinian Authority and Jordan.
Source: Israeli Airports Authority > Land Borders > Allenby > About