But their assimilation into what is now the fastest-growing Jewish community in the world has been problematic, largely due to linguistic and cultural differences, including varying approaches to defining Jewishness and even sometimes a lack of knowledge about the Holocaust.
The Israeli embassy in Berlin has confirmed that two emissaries from Nativ are due to arrive in Berlin "within the next few weeks" to start their work, which includes trying to encourage Jewish immigration among the Jews from the former Soviet Union and offering practical help with their move to Israel. Specific details have not been made public.
The German press has referred to the operation as being "James Bond-like".
The Central Council of Jews, the political organisation representing German Jewry in Germany, has expressed its fury that Israel is trying to steal its members.
In a recent letter to Olmert, the Central Council described the decision as a "sign of mistrust which we find personally offensive".
Stephan Kramer, the Council's general secretary, said it gave the impression that the "connection to Israel" and "imparting Jewish values" was not important to Jewish community leaders, when it belonged to the "foundations of Jewish existence".
He said: "If you read the cabinet decision, you get the impression that Jewish Germans need to be evacuated," he said. "That sends a fatal signal."
The German government has joined the row, insisting it should be up to individuals to decide where they live.
"Whether someone wants to live in Germany or Israel is a decision that only they can reach themselves," said the German foreign minister, Frank-Walter Steinmeier, on a recent visit to Israel.
Nativ was set up in the 1950s as an independent intelligence agency to build up contact with Jewish activists in the Soviet Union and to encourage Jewish immigration or "aliya" to the state of Israel.
But after the collapse of the Soviet bloc, the organisation lost its raison d'etre, until earlier this year when Avigdor Liebermann, Israeli's strategic affairs minister, became its new head and was given the task of leading the so-called Operation Germany on a budget of £1m a year.
Liebermann has accused the Central Council of Jews of poorly representing the 100,000 Jews from the former Soviet Union who are registered with it and favouring instead veteran German Jews who make up only a small portion of Jews in Germany.
"There's a huge potential here for expanding aliya," he has said.
The central council and its welfare arm, the ZWST, have said they fear the existence of Nativ will further deepen the rift between the older community and the former Soviet Jews at a time when the revival of Jewish life in Germany is being widely celebrated.
New synagogues are being built and painstakingly renovated around the country, such as the Ryke Strasse synagogue in Berlin
But there are many rows about "how Jewish" members are.
"Often when there are arguments you hear the cry: 'you're not a proper Jew'," said Moishe Waks of the Ryke Strasse synagogue and a Jewish leader. "Even though they would have been persecuted under Nazi racial laws, many immigrants from the Soviet Union are not considered to be proper Jews by the rest of the community because their mothers were not Jewish."