The Israeli official, speaking on condition of anonymity, confirmed the meeting was cancelled after Gabriel decided to go ahead with talks with Israeli rights groups Breaking The Silence and B'Tselem.
Breaking The Silence seeks to document alleged Israeli military abuses in the Palestinian territories, while B'Tselem has worked on a range of issues and has strongly opposed Israeli settlement building.
Cancelling the meeting between Netanyahu and Gabriel was a rare step, but in line with the current right-wing Israeli government's stance against groups it accuses of having political agendas.
Gabriel earlier told journalists in Ramallah, where he held talks with Palestinian Prime Minister Rami Hamdallah, that he had still hoped to meet Netanyahu and the rights groups.
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He had also told German public television station ZDF that a decision to cancel the meeting would be "extremely regrettable".
"It is completely normal that we speak with civil society representatives during a visit abroad," he said.
Gabriel added that it would be "unthinkable" to cancel a meeting with Netanyahu if he met critics of the German government during a visit to Germany.
The two NGOs are due to hold a joint meeting with Gabriel on Tuesday evening, a source from one of them said.
Such disputes have arisen in the past between visiting foreign officials and Israel's government.
In February, Israel reprimanded the Belgian ambassador after the country's prime minister, Charles Michel, met with B'Tselem and Breaking The Silence during a visit to Israel.
However, there was no public rebuke from the government when British Foreign Minister Boris Johnson met with anti-settlement NGO Peace Now during a visit in March.
Israel has occupied the West Bank for 50 years and Jewish settlement building in the Palestinian territory has drawn intense international criticism.
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Israeli settlements are seen as illegal under international law and major stumbling blocks to peace efforts as they are built on land the Palestinians see as part of their future state.
Germany has been among critics of Israeli settlement policy.
A German government spokesman said in February that a summit with Israel planned for May had been delayed, with Israeli media reporting it was due to the Jewish state's controversial new settlements law.
Israel passed a law in February that legalises thousands of settler homes built on private Palestinian land in the West Bank.
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