The threat, received in the morning, referenced an attack on the Tehran-Hamburg flight, police said.
They did not give any further details but said they were taking it seriously.
"The federal police this morning received by email a threat of attack against an Iranian aeroplane," a police spokesman said.
The German air force said its planes were scrambled to accompany the aircraft from where it entered German airspace, east of Berlin, to Hamburg.
The jet from Tehran with 198 passengers and 16 crew members landed safely at Hamburg airport at about 12.20pm.
The passengers and crew underwent a security check, and the aircraft and luggage on board was searched.
A spokesperson for the airport described the measures, which included interviewing passengers, as normal procedure when a threat is classified as serious.
There was no immediate update on the outcome of those checks.
No take-offs or landings at the airport in northern Germany had taken place since 12.40pm local time, according to German news agency dpa.
But traffic had resumed 90 minutes later, around 2pm.
"Flight service has resumed," the airport said on its website. "There may still be flight delays."
The news came on the first day of a special meeting of the German and French governments in Hamburg, with Chancellor Olaf Scholz and President Emmanuel Macron both attending.