ITV News' political correspondent, Carl Dinnen, reports.
Theresa May has said Britain needs more than a "Brexit prime minister" as she dismissed suggestions the next Tory leader must have supported a vote to leave the European Union.
In her first television interview since announcing her candidacy, the home secretary told Peston on Sunday that she would bring the Remain and Leave sides together and build "a government for all the people".
Michael Gove, Andrea Leadsom and Liam Fox, her rivals for the leadership, have said only someone who backed Brexit should replace David Cameron.
May, who supported the Remain camp, is the overwhelming favourite to become prime minister but she insisted she is not taking "anything for granted".
She said there had to be a "proper" leadership contest and not a coronation to ensure party members have their say, amid suggestions her rivals should back out if she wins large support from MPs in the early rounds of voting.
She also ruled out holding an early general election if she won, saying it would be "destabilising".
May said Britain needed to quickly establish its negotiating position with the EU before invoking Article 50 - the clause which triggers the UK's exit from the bloc - immediately.
She said all efforts must be made to achieve the right deal on free movement and trade.
May said there was "no single silver bullet that solves peoples' concerns about immigration" but said she believed she could reduce net migration down to what she called "sustainable levels".
She said the public, in its decision to vote for Brexit, had delivered a "very clear message that we cannot allow free movement to carry on" as it is.
She was also criticised for failing to confirm whether EU nationals living in the UK will be able to stay after Brexit.
She told Robert Peston that it would be subject to "negotiations".
"Of course, as part of the negotiations we will need to look at this question of people who are here in the UK from the EU," May said.
She was immediately condemned by former shadow home secretary Yvette Cooper who said the comments "were not good enough".
- Watch Theresa May's full interview with Robert Peston