Mr Lidington was speaking shortly before cross-party Brexit talks with Labour resumed in Whitehall.
According to the Telegraph, Theresa May reportedly entered secret discussions over the possibility of staging a second Brexit referendum, the newspaper said on Sunday.
Theresa May's deadline for Britain leaving the European Union has slipped again after the government set a fresh target for her Brexit blueprint to be approved by MPs.
No confidence vote in Parliament: The PM would have to resign if she lost a confidence vote in Parliament.
Tuesday's meeting with Sir Graham Brady comes after the 1922 Committee requested "clarity" about the Prime Minister's timetable for standing down and triggering a leadership contest.
De facto deputy prime minister David Lidington confirmed the elections but said the government was "redoubling" efforts to get a withdrawal agreement through by July, so that elected MEPs don't need to take their seat.
Conservative Party chiefs have signed off a cut-price campaign for European parliament elections after finally admitting that the polls will go ahead two weeks tomorrow.
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Highly anticipated: All signs suggest Google is ready to launch cheaper versions of its Pixel 3 at Google I/O next week. There was also the wildest feature of all: using Assistant to make restaurant reservations with Google Duplex.
Mrs May's hopes of rapid progress faded as cross-party talks with Labour on the potential shape of a Brexit deal entered their sixth week.
"She needs to be given a date".
Pro-EU Labour voters could turn to Change UK, another new party, or support the Liberal Democrats.
"And then, whoever is leader of the Conservative Party can then lay out their stall as to the next instalment of negotiations".
Both the main parties are paying the price for failing to deliver Brexit.
Labour sources were dismissive of weekend reports the government would offer new concessions, including a temporary customs union with the EU until a national election due in June 2022, saying such a proposal would not go far enough. Some members - even some who want May to resign - are nervous that it could look bad to alter party procedures exclusively with a view to ousting the prime minister.