The call came after the Electoral Commission raised concerns about “troubling” claims of people casting votes in more than one location in the June 8 poll.
The elections watchdog said it had not found evidence of widespread abuse, but called on the Government to consider ways of cutting the risk of abuse of voting rules.
Voters are allowed to be registered in two areas, and many students are listed on electoral rolls both at their home and university addresses.
But it is illegal to vote more than once in a general election or other national vote.
Asked whether Theresa May thought anyone caught voting twice should be prosecuted, the Prime Minister’s official spokesman said: “Yes. One-person-one-vote is a core principle of the democratic process in this country. It’s vital this is upheld.”
He added: “It is illegal to vote in more than one location in a general election or in other national polls, such as a referendum.
“It’s clearly explained on the poll card. Conviction carries a financial penalty.
“If anyone has evidence of an individual voting twice at the same election, they should report it to the local returning officer and to the police.”
The Electoral Commission said in a report on Monday that it had received more than 1,000 emails from members of the public, along with 38 letters from MPs, raising the issue of double-voting.
The watchdog revealed it is working with police on how to investigate allegations.
Its report stated: “Although people may lawfully be registered to vote in more than one place in certain circumstances, it is troubling that some voters appear to have admitted voting more than once at the general election, which is an offence.”
An increase in the number of younger voters is claimed to have fuelled the surge in support for Jeremy Corbyn’s Labour Party.
Nearly two million young people applied to vote after Mrs May announced she planned to call a snap election, according to the Electoral Commission.
Around 500,000 more people were registered for the June 8 poll than at the 2015 general election, taking the electorate to 46.8 million, the largest so far.