The present rules state that the Conservative Parliamentary Party will select (from their own membership) two candidates that will then be voted on by the Conservative Party's general membership. For the record, it doesn't say that they have to be willing(!).
The current rules for electing the leader of the Conservative Party were
introduced in 1998. The system gives every member of the Conservative Party a say in the election of the leader. In brief, the election system consists of two stages:
• Conservative Members of Parliament select a choice of two
candidates to present to the membership of the whole Party
• Party members vote, on a "one member one vote" basis, for their
preferred candidate from a shortlist of two
HoC Briefing Paper: Leadership elections: Conservative Party
It is simply inconceivable, given that there are potentially 315 candidates for the role, several of whom have stood in prior elections or expressed that they would like to become leader, that no two members would be interested in becoming leader of the party. In extremis it can be assumed that pressure would be brought to bear by the Whips Office, the 1922 Committee and the Chairpeople of the MP's own local associations to essentially force some members to stand.
The Constitution of the Conservative Party makes allowance for a change to the rules for the election of leaders. This requires only a simple majority of the Executive Board of the 1922 Committee. They could, for example, allow non-MPs to offer themselves for the leadership (such as the 248 Conservative Peers of the Realm, MSPs or AMs).
Upon the initiation of an election for the Leader, it shall be the
duty of the 1922 Committee to present to the Party, as soon as
reasonably practicable, a choice of candidates for election as Leader.
The rules for deciding the procedure by which the 1922 Committee
selects candidates for submission for election shall be determined by
the Executive Committee of the 1922 Committee after consultation of
The 1922 committee could then nominate a single candidate who would then immediately become leader (although this would be politically unpalatable)
In the event of there being only one valid nomination at the close of
nominations prior to the first ballot being held by the Parliamentary
Party for the election of the new Leader, the election of the nominee
may if so ordered by the Board be ratified by a ballot of the Party
Members and Scottish Party Members to be held within one month of the
close of nomination.