William Spindler, a spokesman for the United Nations refugee agency in Geneva, said there was no agreement in place with other countries to take any of them. He said the agency would need countries to come forward with offers to take them, but added, “We are confident we will be able to find places for these 16,000 people.”
Such promises can be hard to keep. The European Union is still struggling to get its member countries to live up to a 2015 agreement to take migrants from the Middle East, Africa and Asia.
When the Israeli government announced the plane ticket-or-prison policy three months ago, it caused a popular outcry, with many Israelis arguing that forced deportation did not sit well with Jewish values. Newspapers filled with petitions signed by Israeli doctors, pilots, rabbis, musicians and Holocaust survivors, all opposing the policy.
Stickers appeared in pubs, cafes and other establishments saying, “This establishment opposes the expulsion of asylum seekers,” and citing biblical verses like Leviticus 19:34: “The stranger who resides among you shall be to you as one of your citizens; you shall love him as yourself, for you were strangers in the land of Egypt.”
But the agreement to let many of them stay in Israel drew harsh criticism from some of Mr. Netanyahu’s right-wing coalition allies. Naftali Bennett, the education minister and leader of the far-right Jewish Home party, said the deal would “turn Israel into an infiltrator’s heaven.”
The asylum seekers are concentrated in a few places, notably south Tel Aviv, and the Israeli government and the United Nations refugee agency said they would encourage a more balanced distribution of the asylum seekers who remain in Israel. That effort will include vocational training and job placement, and will offer help integrating into new communities.
Israel has never disclosed the names of the African third country or countries with which it said it reached a secret agreement to take the asylum seekers. The migrants, themselves, said that under a previous effort that paid migrants to leave, the main destination was Rwanda, with others going to Uganda.
But migrant and human rights organizations said many of those people did not find work or receive legal status at their destinations, and many continued on their journeys, often putting themselves in danger. The Rwandan government also denied having signed any secret deal with Israel and said it would only accept migrants who came voluntarily.
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“It became clear that this third country was not upholding the conditions, not withstanding the pressure,” Mr. Netanyahu said. “So from the moment it became clear, in recent weeks, that the option of the third country no longer existed, we were in a catch, meaning they would all have remained here.”
In addition, in response to petitions by Israeli human rights advocates, Israel’s High Court of Justice issued an emergency injunction last month, instructing the state to suspend its plan to begin deporting single adult men on April 1.
The issue of the African asylum seekers has long roiled Israeli society. From 2005 until 2012 about 60,000 surreptitiously crossed into Israel over the once-porus border with Egypt, most of them Sudanese or Eritreans who, under international conventions, could not be sent back to their home countries, where they could face persecution.
Anger built up among many of the veteran residents of south Tel Aviv, who complained that the migrants brought crime and made their neighborhood unlivable. Other Israelis volunteered to help the migrants. Many passed through an open but remote detention center in the Negev desert.
The government stopped the flow by building a steel barrier along the border and by making the lives of those already in Israel increasingly uncertain and uncomfortable. Very few of those who filed asylum requests have received refugee status. Government ministers routinely refer to them as “infiltrators” and describe them as economic migrants who threaten Israel’s future as a Jewish state.
At least 20,000 of the migrants have left Israel, and Mr. Netanyahu said he had made it his mission to deport the rest.
Susan Silverman, a rabbi and the sister of American comedian Sarah Silverman, conceived and co-founded an initiative called Miklat Israel, Hebrew for Israel Sanctuary, which signed up hundreds of families from around the country willing to adopt asylum seekers and hide them in their homes, if necessary.
“For such a little country, and with all of our faults, boy are we scrappy,” Rabbi Silverman said, referring to the struggle to halt the forced deportations. “So many people cared about the fate of the asylum seekers in Israel.”
Several migrants and their allies gathered in Rabbi Silverman’s Jerusalem home on Monday to watch the televised news conference announcing the new agreement.
Berhan Nagasi, 32, a leader of the struggle, was among them.
“We worked 24 hours a day,” said Mr. Nagasi, an Eritrean who arrived in Israel 12 years ago. He said the new deal made him both happy and full of sadness.
“Israel could have absorbed all the people,” he said of the asylum seekers, speaking in fluent Hebrew. “Now they will have to begin from zero, in another culture, and get to know another country.”