About 12,000 civilian employees who work at US military installations in Germany could be affected, the Verdi union warned.
"Verdi is calling on the federal government to assume financial responsibility for local staff here," the union said in a statement.
Previous shutdowns had not led to workers' pay being interrupted, Verdi deputy leader Christine Behle said, but she accused the US government of this time "apparently not being ready to follow German law".
"Salary interruptions due to budget disputes are not permissible under German law," she said.
"We support any political solution that avoids interruptions to salary," she added.
A spokeswoman for the German finance ministry said it "remains unclear" whether the current shutdown will affect civilian employees working for the US armed forces in Germany and it was "monitoring further developments".
"In the past, these civilian employees... were exempt from a (threatened) shutdown and received their wages," it said.
The US army and embassy in Germany did not immediately respond to requests for comment.
Germany hosts tens of thousands of US military personnel locations including the Ramstein Air Base, which serves as a US Air Force headquarters for Europe and Africa.
A spokeswoman for the interior ministry in the state of Rhineland-Palatinate, where some 6,300 of the civilian workers are based, said that "as of today, the deadlines for timely wage payments have not yet expired".
At 20 days, the United States is enduring its longest full government shutdown ever -- the third-longest if partial stoppages are included -- in a standoff over healthcare subsidies.
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