The conservative Jomhuri Eslami reports the couples and another individual were running a website, Iran Multiplication, which was aimed at promoting illicit sexual relations.
The couples were said to have carried out sexual acts in the presence of each other and several times with multiple partners, the report added.
Those arrested held university degrees, while some were government employees and had children.
The paper gave no further details about their identities or when or where they were arrested.
Adulterers face stoning
Extra-marital sex is illegal in Iran where Islamic sharia law is the principal source of legislation.
If found guilty of adultery, those arrested in the crackdown face being stoned to death.
Reports of partner-swapping are a rarity in conservative Iran, but in March the elite Revolutionary Guards said it had launched a crackdown on several groups who had set up anti-Islamic and pornographic Internet sites.
The Guards, set up to defend the ideals of the Islamic republic, said they has "dismantled several networks that had set up anti-religious, anti-revolution and obscene Internet sites".
Among those also targeted are people deemed to be Satan worshippers, while moral police often also carry out raids on concerts and parties as part of their tough crackdown on "un-Islamic" attire and behaviour.
Iran's moral police have arrested a dozen couples for engaging in illicit sexual acts, including swapping of partners.