The government has indicated that Britain could still leave the European Union before those elections if the two main parties could reach a compromise on a Brexit agreement in the next couple of weeks.
May's authority has been shattered by her three-time failure to get an exit deal approved by parliament and a pledge to quit once Brexit is delivered, driving speculation about her successor and a possible national election.
However, there is no formal mechanism with which dissatisfied lawmakers can oust her without also raising the possibility of a general election and a Labour government.
The Observer newspaper in London reported Sunday that the latest Opinium poll suggested the Conservatives have dropped six points over the past fortnight, giving Labour a seven-point lead. He described the delay to Brexit as "political death" and urged May to avoid taking part in the European elections.
The Telegraph commented that May now faces mounting calls by MPs and senior grassroots figures to name a date for her departure as it appeared Britain would have to participate next month in elections to the European Parliament.More news: French press claim Paul Pogba wants to leave Manchester United
"I think what matters is that we have a cabinet that believes in Brexit, and we believe in Brexit, whichever way we voted in the referendum, because we're all democrats and we think that there are great opportunities for this country, whichever choice it makes", Hunt told BBC Radio 4's Today programme from Japan, where he met the Japanese prime minister, Shinzo Abe.
Lidington said while there was no deadline, both sides would take stock in 10 days.
Lidington said: "These are hard, delicate negotiations where we are testing out other people's ideas".
Britain's European Union membership is due to end on October 31, with or without a deal.
Last week the European Council extended Britain's membership of the bloc until October 31 to give more time for the British parliament to agree on a withdrawal deal.