Hampshire Police said 35 people across the county had surrendered their passports with four more people retuning them once back from holiday.
Officers will be posted at Portsmouth port in a bid to stop anyone evading the ban and sneaking on to ferries ahead of the tournament.
The force would not disclose whether its Football Unit officers would be travelling abroad to keep an eye on fans.
Superintendent Andy Houghton, head of operations for Hampshire and Thames Valley, said: ‘We have enhanced patrols in Hampshire for the England games and a few of the Poland fixtures and they will be supported by officers already working in their local communities.
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‘We have worked with licensed premises in advance of the tournament, providing them with best practice and guidance and ensuring they are fully prepared for the competition.
‘We will be seeking to ensure that the premises are operated in a safe and responsible way throughout the event and are encouraging venues to work with us should any issues arise.’
The tournament starts in Paris tomorrow night.
Separately, Action Fraud has warned people fraudsters are selling fake tickets to those buying last minute.