The plan, revealed by Police Scotland, follows an increasingly tense and hostile election campaign that has seen clashes between far-right activists and anti-racism protesters.
Deputy Chief Constable Will Kerr, of Police Scotland, said four units comprising about 100 public order officers would be "strategically placed" throughout the day.
He said: "We would never normally need that sort of support, particularly for European elections.
"We do now. The tone is just fundamentally different."
Police in London said they would be revealing any extra measures in an announcement later on Thursday.
In a sign of how seriously police are treating threats of violence linked to the EU election, Sadiq Khan was given round-the-clock protection after receiving a deluge of death threats in the lead-up to polling day.
The London mayor said his Muslim faith had made him a target in the wake of the 2016 referendum.
At the weekend, officers warned there would be arrests after two police vehicles were damaged during a disturbance at a Tommy Robinson campaign event.
Footage posted on social media showed clashes between supporters of the campaigner, whose real name is Stephen Yaxley-Lennon, and counter-demonstrators in Oldham, Greater Manchester.
Policing plans for previous elections have tended to focus on preventing voter intimidation and fraud.
During the last European elections in 2014, the government was advised to station police in polling stations in 16 areas at risk of vote-rigging.
At the time, the Electoral Commission raised concerns about voter impersonation in Birmingham, where a judge condemned activities which he said would disgrace a “banana republic” following the exposure in 2004 of systemic postal voting fraud in two wards.
It also named the London borough of Tower Hamlets, where a reporter for The Independent was beaten up while investigating allegations of voting irregularities during the 2010 general election campaign.
And during the 2017 general election, armed police were deployed to some polling stations; however, this came at a time of heightened tensions following three terror attacks in less than three months.