Turkey's NTV television said the attacker tried to enter the well-defended embassy but was "neutralised" by being shot in the leg.
Employees of the embassy took refuge in a shelter, it added.
Israel's foreign ministry said the assailant tried to attack the Israeli embassy but was shot before reaching the building.
"The team members are unharmed. The assailant was injured before he reached the building," a ministry statement said.
The health condition of the attacker was not made clear. It was also not clear if the person had been shot by Turkish security forces or embassy guards.
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CNN-Turk television said a suspicious package was also being investigated in front of the embassy.
The attack comes three months after Turkey and Israel signed a deal to restore their ties which hit an all-time low after the 2010 raid by Israeli commandos on a Gaza-bound Turkish aid ship that left 10 Turks dead.
Under the deal, Turkey and Israel are to begin the process of exchanging ambassadors to fully restore their diplomatic ties although this has yet to formally take place.
Israel had already offered compensation and an apology over the raid, but with the agreement it also eased the naval blockade on the Hamas-controlled Gaza Strip, allowing Ankara to deliver humanitarian aid to Palestinians there.
The incident also comes with foreign missions in Turkey on a state of high alert following a spate of attacks across the country this year blamed on jihadists and Kurdish militants.
The British embassy in the Turkish capital Ankara was closed on Friday over security concerns while the German embassy offered only limited services.