The breach was uncovered by the Government Communications Headquarters (GCHQ), Mirror online reported.
An investigation by the intelligence organisation has discovered that IS extremists have been targeting information held by some of Prime Minister David Cameron's most senior ministers, including Theresa May, the Home Secretary.
The extremists could have discovered events that government figures and members of the Royal family were expected to attend as a result of the hacking. the GCHQ said.
It is understood that at least one of the plot's ringleaders was killed by a drone strike in an operation disclosed by Cameron earlier this week.
It is still unclear what information the extremists were able to access. However, officials were told to tighten security procedures, including changing passwords.
The disclosure shows how the so-called "war on terror" has developed because of IS' cyber capabilities - computer hackers are known to have been recruited by the group to "actively target the West".
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