German broadcaster RBB found up to 700 cases in Berlin alone. "There are many unreported cases," said Ole Schröder, a top interior ministry official.
There has been a rise in asylum requests from pregnant women from Vietnam, Africa and Eastern Europe.
A new law is being drafted to tackle the fake paternity racket, RBB reports.
Some pregnant immigrant women are reported to have paid fake fathers and solicitors as much as €5,000 (£4,356; $5,628) to get paternity registered. Once that is done, the baby automatically becomes a German citizen and the mother has the right to stay.
However, in 2013 a German Constitutional Court ruling said that even in suspicious cases it might not be worth contesting paternity, because the child could end up stateless and left in legal limbo.
It is generally easier for refugees fleeing conflict to get residency in Germany than for economic migrants who have escaped poverty in Africa, Asia or Eastern Europe.
Germany has moved to tighten asylum rules since the 2015 influx of more than 800,000 migrants, most of them refugees from Syria, Iraq and Afghanistan.
Immigration is a major issue ahead of parliamentary elections in September. The influx hit some local authorities especially hard, as they struggled to house and integrate the newcomers.
However, there was a sharp drop in the numbers claiming asylum in 2016, after barriers were erected on the Balkan migration route.
Massive drop in German asylum seekers
Record migrant numbers leave Germany
Ten attacks a day on Germany migrants
Polls suggest that the nationalist Alternative for Germany (AfD), opposed to mass immigration, has gained support and is on course to enter the federal parliament (Bundestag) for the first time.
Mr Schröder, a member of Chancellor Angela Merkel's centre-right Christian Democrats (CDU), said "we have a lot of evidence from the immigration authorities - fake fathers are making money out of this... in other words, this involves a considerable amount of criminality".
In one case reported by RBB, a far-right sympathiser, convicted for sporting neo-Nazi symbols, claimed to be the father of a Vietnamese child.
Martin Steltner, a state prosecutor in Berlin, told ARD television there were signs of widespread abuse involving paternity, for example one man claiming to be the father of 10 children.
The fake fathers rarely pay any child support, as many are living off social welfare, ARD reports.
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, whom governments are likely to rule are economic migrants.
German officials say growing numbers of pregnant immigrant women are paying German men to pose as fathers so that they can qualify for residency.