The app, which was set up by an Estonian-based web company, allows users to give "sincere and honest feedback" about others anonymously.
It has about 30million users worldwide, and while it is supposed to be for over-18s, youngsters are able to bypass the restriction by lying about their age.
Users are given a personal web address to display on their Instagram or Snapchat feed and invite anonymous comments, which they can then publish or keep secret.
One girl recently received the comment: "You are a bitch, if you don't change then it's not just me who will want to kill you."
This month, a letter sent to parents by Wrotham School in Sevenoaks, Kent, warned: "This site is often misused as a way of making offensive, nasty, rude and inappropriate comments.
"Young people may be using these sites without your knowledge and they might be receiving incredibly offensive comments that could lead to them becoming seriously distressed and upset."
And the Rawlett School in Tamworth, Staffordshire, posted online: "Students are using the app to carry out cyberbullying. This is happening not just between students at Rawlett but between students across other schools."
Sayat.me has allegedly been linked to a number of suicides, and a petition set up by a 13-year-old girl to urge Theresa May to shut it down now has 1800 signatures.
Millie Clatworthy, the daughter of a British Airways captain from Maidenhead in Berkshire, said: "I started the petition because I saw this app cropped up over social media and I saw a lot of my friends were using it as well.
"I knew it was a horrible website. Because it's anonymous, it will lead to online bullying and things like trolling and harassment."
A post on the petition from a member of George Hessay's family blamed the app for his death.
The teenager, from Rawcliffe, East Yorkshire, who died on May 10, was described as the class joker but he had been bullied online, his family told friends. He is thought to have hanged himself.
Dan Raisbeck, of The Cybersmile Foundation, an anti-cyberbullying organisation, said the UK authorities are powerless to act because the site is hosted abroad.
The Estonian-based software company behind Sayat.me is listed as Mooncascade LLC. It was co-founded by IT consultant Asko Seeba.
In a statement, chief executive Talving said: "We have been made aware by police that they are investigating a bullying-related suicide and we offer our sincere condolences to those affected by this loss.
"We deplore bullying and want it to have no place on our site. We will offer any assistance we can to the police. We have suspended use of the website to show how seriously we take these matters."
Where to get help:
• Lifeline: 0800 543 354 (available 24/7)
• Suicide Crisis Helpline: 0508 828 865 (0508 TAUTOKO) (available 24/7)
• Youthline: 0800 376 633
• Kidsline: 0800 543 754 (available 24/7)
• Whatsup: 0800 942 8787 (1pm to 11pm)
• Depression helpline: 0800 111 757 (available 24/7)
• Rainbow Youth: (09) 376 4155
• Samaritans 0800 726 666
• If it is an emergency and you feel like you or someone else is at risk, call 111.
