The distinctive icon is already on some web services children use
The Child Exploitation and Online Protection centre (CEOP) wants the sites to install its "report abuse" button that connects people to police.
CEOP research shows some sex offenders are starting to use social network sites, such as MySpace, Bebo and Facebook, to seek out victims.
Jim Gamble, head of CEOP, says: "The more children go on social networks, the more offenders follow them."
The button lets people report instances when they suffer or witness inappropriate sexual contact. In August 2006 Microsoft agreed to put the distinctive red icon on the of its instant messenger service - MSN Messenger.
CEOP research shows that, while chatrooms and instant messaging services are the main places sex offenders go in search of victims, social networks are attracting them too.
Latest figures show that around one million children under 16 use Bebo, while 600,000 minors are on MySpace.
The networking sites say they make it simple for users to report abuse, though those reports usually go to the site administrators rather than the authorities.
In the US social network sites are under pressure from politicians who want them to do more to protect their most vulnerable users.
American law enforcement agencies are also seeking to make the social network sites introduce checks to prevent under-age users from coming to the sites and to keep sexual predators at bay.
All the networks make new users reveal their age when they sign up, but there is disagreement about whether any new system of age verification would be effective.
MySpace says anyone under 14 is banned from using its service.
A spokesman said: "MySpace has developed search methodology and algorithms to seek out underage users, relying on several thousand terms commonly used by under-age users to identify them and delete their profiles."
MySpace and other networks also scan profiles for clues that users are older than they claim to be.
At Greenford High School in West London students have made a film about social networking and its dangers, with scenarios illustrating how easily young people can be tricked.
It seems to have worked. All the pupils we spoke to said they would never accept anyone as a "friend" on Facebook or Bebo unless they knew them.