"We have got the Brexit Party and UKIP suggesting people should vote for them when they don't have a single MP with which to deliver Brexit, which means they are baldly incapable of doing so".
Graham Brady, chairman of the 1922 Committee which can make or break party leaders, said May told him she was determined to get her Brexit deal ratified as soon as possible which would pave the way for her departure.
Brexiteer Ms Jenkyns told Mrs May she had "failed" in European Union withdrawal negotiations and forfeited the trust of the public, but the Prime Minister dismissed her suggestion.
'We're going to give it our best shot but that's the best I can say at this stage.
She is immune from a leadership contest until December after surviving one previous year.
Last week, May had also made bullish efforts to make opposition Labour Party leader Jeremy Corbyn agree on a cross-party deal for ensuring UK's smooth exit from the EU.
Mrs May retorted: "This is not an issue about me and it's not an issue about her".
Earlier this week, UK Cabinet Office Minister David Lidington confirmed that the United Kingdom would be unable to ratify the Brexit deal in the next two weeks, and would organize European Parliamentary elections on 23 May, despite hopes to push through a Brexit agreement before this date.
The British Green Party on Wednesday launched its European Parliament elections campaign while promising to be "tough on Brexit".
Downing Street said it followed a second day of "extended" discussions in Whitehall which "demonstrates the seriousness with which both sides are approaching these talks".
The two parties have been in discussions for over a month to try to broker a Brexit deal that can secure majority support in parliament after May's minority government suffered three defeats on her preferred deal this year and was forced to delay Britain's original departure from the European Union from March 29 to April 12.