Mohammed Sadique Khan, also known as Sidique Khan, a father-of-one from Dewsbury, West Yorkshire, is alleged to have blown himself up in the blast at Edgware Road last Thursday.
He was one of four men alleged to have been behind the blasts on three Underground trains and a bus which killed at least 52 people.
The TES understands that Khan, who had a 14-month-old baby, may have run a breakfast club at a primary school in Leeds. His wife, Hasina, was also said by neighbours to work in education.
Margaret Morrissey, of the National Confederation of Parent Teacher Associations, said: "This beggars belief. It does not make sense that someone who works trying to help children would go out and remorselessly kill."
Bushra Nasir, head of Plashet secondary school, in Newham, east London, and president of the Muslim Teachers' Association, said Khan was not known to the association. She said: "Our staff and students have been very upset by the attacks."
One of Khan's neighbours, Imran Zaman, described him as a "quiet person".
"I never knew he had a religious background. I go to the local mosque and have never seen him there."
The National Union of Teachers has warned staff to be prepared for increased ethnic tension between pupils. Ethnic-minority teachers may also be vulnerable, the union warned.
The union is sending guidance to schools on how to deal with repercussions from the bombings.