The home secretary promised to bring the Remain and Leave sides together and "govern for the whole country".
But Andrea Leadsom and Michael Gove both said the winning candidate must have backed Brexit.
Candidates have been setting out their stalls in a series of interviews.
Mr Gove, the justice secretary, defended his tactics in the face of criticism, telling Andrew Marr it would have been a "betrayal of this country" if he had allowed Boris Johnson to run.
