Journalist Rachel Johnson - sister of Boris - and ex-BBC correspondent Gavin Esler are among the candidates.
The government says discussions have been "difficult", while Labour says "fundamental issues" remain.
Teenage climate activist Greta Thunberg tells UK politicians a generation's future has been "stolen".
Karen Bradley made the announcement while making a statement about the killing of Lyra McKee.
The US president and Melania Trump will be guests of the Queen and attend a D-Day event in Portsmouth.