Martin Hewitt, who this week became chairman of the National Police Chiefs’ Council, said that public figures should be wary of “consequences that weren’t intended” when speaking about the issue.
Addressing journalists during a briefing at the National Police Co-ordination Centre, he said: “This is highly emotive as an issue as we all know and clearly everyone will have their opinions.
“But I think there is a responsibility on those individuals that have a platform and have a voice, to communicate in a way that is temperate and is not in any way going to inflame people’s views.”
The political situation over Brexit has already seen heated protests on both sides, and a number of MPs have requested beefed-up security in the past six months.
Police have also seen an increase in abuse aimed at politicians, Commander Adrian Usher, who leads parliamentary policing, said.
Mr Hewitt went on: “We are in an incredibly febrile atmosphere as a result of the whole EU exit scenario.
“We know and we can see that people are expressing themselves, we’re seeing a degree of protest, a degree of demonstration, there is a lot of angry talk that you can pick up if you look across social media.
“In any scenario like that where there are a range of outcomes it’s incumbent on anybody in a position of responsibility, and who has a voice, to just think carefully about the way that they express their views and their opinions, so that what they’re not doing is inciting behaviour or causing anybody to behave in a way that we wouldn’t want them to behave.”
He added: “We are in a febrile atmosphere and if you are in a position where you know you are going to be listened to, you need to think very carefully about the language that you are using so that it doesn’t end up with consequences that weren’t intended.”
The number of crimes linked to Brexit, although small, has more than doubled in a fortnight, with 26 last week and 11 the week before.
About half were malicious communications, while the remainder included verbal abuse, harassment and protest activity.
One protester admitted climbing on to a station roof and causing widespread disruption to Eurostar services, while British Transport Police are looking for another who planted devices aimed at bringing trains to a halt in Cambridgeshire and Nottinghamshire.
In general terms, levels of hate crime in England and Wales have not yet fallen back to the levels recorded before the 2016 Brexit referendum.
In 2017/18, 94,098 of these offences were recorded, a rise of 17% on the previous year, thought also to be fuelled by the terrorist attacks in London and Manchester.
Chief Constable Charlie Hall, the NPCC lead for operations, said that the police have pushed back on other authorities to make sure that officers are only used “if absolutely necessary”.
This includes issues such as queuing at ports and maintaining the supply chains of food, fuel and medicine.
He said: “We’ve been very clear that policing support should only really be called on if absolutely necessary in dealing with the wider civil contingencies.”
Mr Hall added: “Our push has been back to those sectors, those parts of government, the private sector, to say ‘it’s your responsibility to look at your individual supply chains and you should not be looking to police to come in to supplement and keep your supply chain running’.”
Local resilience forums, which include representatives from councils and the police, have carried out training exercises to deal with shortages, and police training has included dealing with disorder including looting.
Under existing national contingency plans, every police force is expected to make a certain number of officers available to be deployed wherever they are required.
More than 10,000 riot-trained officers can be deployed within 24 hours if needed, with 1,000 available in the first hour. This is more than were mobilised during the 2011 riots.
There are also specialist teams such as dog handlers, armed police and search-trained officers which can be called upon.
Currently 1,000 have received extra training so that they can be deployed to Northern Ireland if needed. The police there use armoured Land Rovers and water cannon, unlike their mainland counterparts.
So far 15 forces have placed restrictions on annual leave, and two have officially asked for mutual aid - Kent, which covers the port of Dover, and Hampshire, which covers Portsmouth.
Forces are expected to cut down on “peripheral police activity” like crime prevention work and to cancel training if needed.
Mr Hall said special constables, volunteer police officers, could also be used if required.