A website of the dreaded terrorist group is encouraging "lone-wolf" attacks on newsreaders alongside the location of their offices, the 'Daily Star' reported.
The newspaper said it has contacted police to alert them about the website and counter-terrorism officers are now looking into the case.
"The Islamic State website posts instructed followers to target BBC and Sky newsreaders and posted the location of their offices," the daily said.
"Terror bosses went on to instruct fanatics to attack popular tourist attractions. They include Downing Street, Big Ben and, chillingly, Westminster, the scene of last month's terror attack," it adds.
Further posts reportedly provided a list of names of British MPs, police stations, government buildings, Army bases, shopping centres and airports.
The "terrifying lists" were posted on an "offshoot" of a well-known terrorism website, the newspaper claims.
The website had pictures of alleged victims of a coalition bombing in Mosul, Iraq, with the message: "By Allah! We will not forget! Even after some time we will never forget!".
Security expert Will Geddes told said that public figures commenting on ISIS should take security seriously.
He said: "Anyone potentially in the public eye talking about IS needs to be reasonably concerned about their safety.
Lists like this provide a paint-by-numbers approach to terrorism. It's about giving supporters ideas on who or what to target."
The UK's security forces are encouraging any online suspected terror-related information to be reported at gov.uk/report-terrorism.