Gemalto produces the SIM cards for cellphone companies, burns an encryption key on to each and sends a copy of the key to the provider so its network can recognise a phone.
According to the Intercept, GCHQ targeted Gemalto employees, scouring their emails to find individuals who might have access to the company's core networks and systems generating the encryption keys.
The goal, the publication said, was to steal large quantities of keys as they were being transmitted between Gemalto and its wireless network providers.
The NSA did not immediately respond to a request for comment.
Stealing the encryption keys makes it possible to eavesdrop on otherwise encrypted communications without undertaking the more difficult challenge of cracking the encryption.
It also avoids alerting the wireless company or the person using the phone.
The NSA's interception of phone calls and other content is bound by different legal standards. A warrant is required to target an American's calls and emails.
In general, targeting a foreigner's communications for collection overseas does not require a warrant.
The publication cited one 2010 GCHQ document that said agency personnel developed "an automated technique with the aim of increasing the volume of keys that can be harvested".
The document acknowledged that in searching for keys, operatives would harvest "a large number of unrelated items" from targeted employees' private communications.
However, it said, "an analyst with good knowledge of the operators involved can perform this trawl regularly and spot the transfer of large batches" of keys.
The GCHQ documents also described operations targeting other major makers of SIM cards, the Intercept said.
Hack attack
• British and American spy agencies accused of hacking into Dutch SIM card maker
• It is alleged they wanted to obtain the encryption keys of mobile phones
• Gemalto, the world's largest manufacturer of SIM cards, makes 2 billion SIM cards a year
• Stealing the encryption keys would make it possible to eavesdrop on otherwise encrypted communications without cracking the encryption