Israeli analysts said that it was unlikely Hamas would choose to start firing rockets again after the devastation Gaza experienced after the recent 11-day conflict, but it was likely the group would launch balloons carrying arson and explosive devices into Israel.
Hamas's warnings appeared to hinge on whether or not the march would pass through the Damascus Gate and into the heart of the Muslim Quarter of the Old City.
Israeli settlers celebrated the anniversary of Jerusalem's "re-unification" after Israel captured East Jerusalem from Jordan in 1967 and annexed it, a move not recognised by most of the worldwide community.
Throngs of mostly young religious men sang, danced and waved flags at the Damascus Gate entrance to the Old City that was cleared of its usual Palestinian crowds.
The new Israeli premier is himself a Jewish nationalist but the coalition he leads also includes centrist and left-wing parties and, for the first time in the country's history, an Arab party.
At the height of the tensions, on May 10, Israeli ultra-nationalists held their annual flag parade. Palestinian protesters repeatedly clashed with Israeli police over restrictions on public gatherings during the Muslim holy month of Ramadan in April and May.
According to Palestinian sources, Israel's air force targeted at least one site east of Khan Yunis in the south of the Gaza Strip, which is home to some two million people.
The iconic Al-Aqsa Mosque compound in the heart of the Old City is Islam's third holiest site and a national symbol for all Palestinians regardless of religion.
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Hamas issued a statement calling on Palestinians to show "valiant resistance" to the march.
The Israel Defense Forces was braced for a resurgence in Gaza fighting and a flare-up of clashes across the West Bank on Tuesday amid Palestinian threats of violence if a contentious march goes ahead by Jewish ultranationalists through parts of Jerusalem's Old City.
The IDF said it struck Hamas military complexes and meeting places for the Khan Younis and Gaza brigades.
The so-called balloon unit, Ibna al-Zuwari, announced Monday that it would resume launching balloon-borne incendiary and explosive devices into southern Israel from Tuesday morning.
Prior to Tuesday's march, Israel beefed up its deployment of the Iron Dome anti-missile system in anticipation of possible rocket attacks from Gaza.
Omer Bar-Lev, the new Cabinet minister who oversees police, said he met with police, military and top security officials to review the plan. He "underscored the need to avoid friction and protect the personal safety of".
United Nations deputy spokesman Farhan Haq said United Nations officials have made clear "the need for all sides to refrain from unilateral steps and provocations, for them to exercise restraint and allow for the necessary work to be done to solidify the current cease-fire".
Prime Minister Mohammad Shtayyeh, of the internationally backed Palestinian Authority in the West Bank, called the march an "aggression against our people".