A copy of the email, seen by the BBC, was signed off in the name of a Merseyside criminal who is serving a prison sentence overseas for hacking and fraud offences.
It did not name any specific target but referred to “schools” in Liverpool.
A statement from Chesterfield said it had received the message in the early hours of Monday morning.
A spokesperson said: “Upon discovery of the email this morning our leadership team took the decision to enter a temporary lockdown procedure, as is national policy, while emergency services were contacted.
“We liaised with officers from Merseyside Police on site who advised our leadership team that they did not believe the communications in question to be credible.”
It said it had resumed normal learning and thanked staff and pupils for their “impeccable behaviour” when lockdown procedures took place.
Ch Insp Phil Thompson said: “We understand that parents and carers will be concerned and as a result we are liaising with all schools and providing advice and reassurance as enquiries continue.
“Based on the information we have received and assessed, there has been no requirement for those schools to close.”
The BBC understands a significant number of parents have taken their children home from some schools.
Sefton Council released a statement urging parents not to collect their children in relation to the email, which it said had been sent to a “very small number” of schools on Merseyside.
“This communication is not believed to be credible, however this has led to understandable concern from some parents,” it said.
“We can confirm once again all schools in Sefton are functioning as normal. Pupils should not be removed from school grounds in relation to this matter.
“Doing so may cause further unnecessary concern and disrupt learning.”