It has been predicted more than 100,000 fans could arrive in a city with a population of less than 40,000.
"We hope the suggested figures are not the same as the actual number of fans who will turn up," Sylvain Robert told the Sport Wales TV programme.
"I don't know how we'd find space for 100,000 people."
Robert says he and the city's officials are "working on all possible scenarios" and a 10,000-capacity fan zone will be set up.
He added: "We are apprehensive about the number of fans who could arrive without a ticket as it's a problem for this type of event."
According to Paul Corkrey of the Welsh branch of the Football Supporters' Federation, Lens' Stade Bollaert-Delelis "is one of the best stadiums in France".
He added: "But the problem is that's all that's good about it.
"The fan zone is very small. To have a tournament of this size there is strange because there are not many amenities there for supporters. I wouldn't go there."
Robert is worried that the town's reputation may be damaged by the tournament.
"It's not that we would be happier to have fewer people, it's that we would be happier if we can host fans in good conditions," said the mayor.
Uefa responded with a statement, which read: "The host city of Lens, in close collaboration with the local authorities and Uefa, will be implementing appropriate measures to ensure that all visitors enjoy matches in a safe and festive atmosphere, be it at the stadium, in the fan zone or in the city centre."
Watch Sport Wales, BBC Two Wales from 19:30 BST on Friday, 6 May and later on demand and on the BBC iPlayer.
The mayor of Lens says he is apprehensive about the number of supporters expected to visit when Wales face England at Euro 2016 on 16 June.