Former foreign secretary and leading Brexiteer Boris Johnson is widely tipped as the most likely to become the next prime minister.
So far, nine Conservative MPs have said they will contest the party leadership.
The race to become Britain's next premier heated up on Sunday as Environment Secretary Michael Gove joined an already crowded field of hopefuls with competing visions of how to finally pull their divided country out of the EU.
"A different deal is, therefore, the only solution - and what I will pursue if I am leader". They're pushing to see Boris Johnson, the figurehead of the 2016 leave campaign, take over and finish what he started.
UK Minister for Environment Michael Gove will offer to provide free UK passports for European Union citizens living in the country at the time of Brexit.
The victor of the competition is expected to be chosen before the end of July, replacing May who is continuing as caretaker prime minister until then.
Meanwhile, Jean-Claude Juncker suggested Britain is drifting towards another Brexit extension in October.
"One of the issues that we have had in the negotiations is that we haven't been able to carry this broad coalition - the coalition that you are going to need to get a deal through Parliament - because we are in a hung-parliament where no party has a majority".
Britain's European Union departure date is now fixed for October 31, although any new leader could ask for a further delay.
- Former Brexit secretary Dominic Raab.
The EU has insisted that unless all three are settled, the bloc will refuse to discuss a future relationship with London, running the risk that the United Kingdom will crash out without a withdrawal agreement in place.
The former Health Secretary warned a no-deal Brexit would topple the Government and cause a general election that could usher in a Jeremy Corbyn-led Labour Government.
The housing minister is credited as the convener of both Conservative leavers and remainers to develop a compromise on May's withdrawal agreement.
As a hardcore Brexiteer, Johnson probably wouldn't mind leaving the European Union without a deal if Brussels refuses to reopen Brexit talks.
Leadsom, who recently quit as House of Commons leader, will run again this time around, with an enhanced reputation as someone who pushed through parliamentary reform and was one of the most outspoken Brexiters round the cabinet table. So far 10 candidates have said they are running.
"Any prime minister would find it very hard to govern if he or she were to pursue a no-deal Brexit without Parliament's permission", he said.
"The DUP do not speak for the people of the north on Brexit and that has been shown once again with the European election results".
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