The pygmy marmosets -- among the world's smallest primates at less than 15 centimetres (6 inches) long -- were born a year ago and had been due to be moved to a Saudi zoo outside the capital Riyadh.
"The Saudi authorities did not grant an import licence for the monkeys... because of the political situation," Jonas Wahlstroem, a zookeeper, at Skansen zoo told AFP.
The refusal to allow the monkeys' transfer comes after Sweden halted a military agreement with the Gulf state.
Riyadh recalled its ambassador to Stockholm last month and froze business visas for Swedish nationals, accusing Stockholm of "flagrant interference" in its affairs.
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Saudi Arabia blamed the dispute on comments by Swedish Foreign Minister Margot Wallstroem who had earlier called the oil-rich state a "dictatorship" that violated women's rights and whipped bloggers.
Swedish officials claimed last week that relations with Riyadh had been "normalised", following intense diplomatic exchanges including letters from King Carl XVI and Prime Minister Stefan Loefven which a high-ranking Swedish MP delivered to Saudi monarch Salman.
"We have been informed that they will start to issue the visas again," foreign ministry spokesman Johan Tegel told AFP.
In the meantime, the monkeys will have to wait their turn in the queue for entry papers to the Gulf kingdom -- along with Swedish business travellers.
"It's almost comical," said Wahlstroem.
"We maybe have to wait until they start re-issuing visas. Maybe the monkeys also need them."