Five senior Tories are in the race to take over from Theresa May, with Matt Hancock being the latest to confirm he will bid for the top job.
Currently health secretary, he confirmed his candidacy to Sky News, saying: "I'm running for the next prime minister because I profoundly believe we need a leader for the future and not just for now.
"We need to deliver Brexit, and I will deliver Brexit, and we need to move this country forward, taking all the opportunities that Brexit brings."
He will take on front-runner Boris Johnson, who has said that under his leadership, the UK will leave the EU with or without a deal.
But Mr Hancock insists a no-deal Brexit is "not an active policy choice that is available to the next prime minister" because MPs have already voted against it.
Joining Mr Hancock and Mr Johnson, the confirmed candidates so far are Foreign Secretary Jeremy Hunt, International Development Secretary Rory Stewart and former work and pensions secretary Esther McVey.
Sky News' political reporter Rob Powell said: "We think there may be as many as 15 that will declare in total.
"That will, of course, be whittled down by Conservative MPs in Westminster to just two... and those will be put to the grassroots Conservative members across the country for them to decide who they want."
Mr Hunt was quick to declare he will campaign after Mrs May tearfully resigned outside 10 Downing Street yesterday.
Mr Stewart told Sky News: "I believe I can get Brexit done... can beat Jeremy Corbyn... and most important of all I believe I can begin to reunify this country."
Adding that he would not serve in cabinet if Mr Johnson was leader, he said: "I sat down with Boris Johnson two weeks ago and he said to me he would not be pushing for a policy of 'no deal' Brexit. I left the room feeling that we had an understanding.
"I now understand, from what he said yesterday, he is going to try to crash us out of the European Union at the end of October... that is not being straight with people."
Work and Pensions Secretary Amber Rudd has ruled herself out of the contest, and Environment Secretary Michael Gove has refused to be drawn on whether he will be entering the race.
When he asked by Sky News if he wanted the job, Mr Gove said: "I think it's really important that all of us recognise that the prime minister has done a great job for this country.
"It was always difficult, and it was moving to see the prime minister speak as she did."
Mr Gove stood in the 2016 leadership race which was won by Mrs May.
Sir Graham Brady quit as the leader of the 1922 Committee - a position which gave him a significant role in the prime minister's departure - in order to consider a leadership bid.
Home Secretary Sajid Javid, a potential leadership candidate, said he had a "frank" discussion with Mrs May about her Brexit deal on Thursday.
He added that "nobody could have worked harder or had a greater sense of public duty".
The new Conservative leader looks set to take over as prime minister at the end of July after Mrs May laid out a timetable for her Downing Street exit.
Former foreign secretary Mr Johnson, who has emerged as the bookies' favourite, stressed he would be prepared to back a no-deal Brexit to ensure the UK leaves the EU on 31 October.
Nominations for the leadership contest will close in the week of 10 June, with MPs involved in a series of votes to whittle down what is set to be a crowded field to a final two contenders.
Tory party members will then decide who wins, which the party hopes will happen by the end of June.
Hustings will take place around the UK, before votes are cast in time for the result before parliament goes to recess in the summer. These could be open to non-members, according to a statement put out by party chairman Brandon Lewis, and Cheryl Gillan, and Charles Walker, vice-chairmen of the 1922 committee.
The UK is still due to leave the European Union on 31 October, but parliament has yet to approve the terms.
Remain-backing Ms Rudd told the Daily Telegraph: "I am conscious the Conservative Party wants someone who they believe is very enthusiastic about Brexit.
"There are all sorts of plans I would like to have when we do leave the European Union but I don't think it is my time at the moment."
The cabinet minister made it clear she would not have a problem working with Mr Johnson in the government again.
Labour leader Jeremy Corbyn said Mrs May's replacement should call an immediate general election.
Despite saying she will leave on 7 June, Mrs May faces a busy fortnight ahead.
US president Donald Trump will make a state visit to the UK and both will then go on to France for the D-Day commemorations on 6 June.
Mr Trump praised the prime minister, saying: "I feel badly for Theresa. I like her very much. She is a good woman. She worked very hard."