- By UTV Deputy Political Editor Tracey Magee
The pair will pitch to Tory party members in the hope of winning their support, as the race to replace Theresa May as leader of the party has reached its final stage.
But here are five questions Mr Johnson and Mr Hunt need to answer on Northern Ireland:
Why did Boris Johnson attend the DUP conference last year and why is Jeremy Hunt so keen to meet the DUP this week?
The Conservatives are a small group of around 500 with no elected representatives in Northern Ireland.
The Confidence and Supply arrangement between the Tories and the DUP has been difficult for local conservatives as they have been fielding candidates in council elections.
There was unease when Boris Johnson spoke at the DUP conference last November and its expected both candidates will questioned on their attitude towards their partners.
The local conservatives have a very different view on liberal issues such as same sex marriage and some feel embarrassed that their party is now depending on the DUP to stay in government.
What proposals do Johnson and Hunt have to break the Brexit deadlock on the Irish border?
This is the key issue in the entire negotiation with Brussels.
But, so far, no one has produced a workable solution.
Local conservatives will want to know what proposals the two candidates have to break the deadlock.
One of the side effects has been a growing call for a border poll which Sinn Fein in particular has been advocating with increasing energy.
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Police chiefs and others have warned any sort of physical border could lead to renewed violence, while others have said it could potentially unravel the Good Friday Agreement.
Both candidates have espoused technical solutions as the answer to border issue but so far no viable proposal has emerged.
The Irish government, backed the European Union, insist no technical solutions presently exist.
Trade experts and various academic studies back up that claim.
Both Mr Hunt and Mr Johnson can expect more rigorous questioning on this matter from an audience with a greater understanding of the border than in Great Britain.
How secure will the union be post Brexit?
There is increasing concern Brexit could lead to the breakup of the United Kingdom.
The Northern Ireland conservatives will want to know what assurances the two candidates can give that the Union is secure.
There is a fear among some that Brexit is being driven by English nationalism and that the views in the rest of the United Kingdom are being ignored.
There is also a concern there could be a domino effect with increasing calls for an independent Scotland and the growing strength of nationalism in Northern Ireland.
Do Johnson and Hunt have any electoral plans?
NI Conservatives, while very keen to see a conservative government continue, are less fixated on the electoral attractiveness of any future leader.
The unique electoral situation in Northern Ireland combined with the fact that the party is so small in Northern Ireland means the performance in elections is nominal.
However members will have questions about the electoral plans any potential leader may have.
They’ll also have questions about the character of the two candidates and concerns about the negative publicity of recent weeks.
What future plans do the candidates have to expand the Tory party?
The party membership will be keen to hear what plans the candidates have for growing the party in Northern Ireland.
The Conservatives have experienced electoral defeat after defeat for many years and are regarded as a lobby group by the local parties and media.
The decline was marked after David Cameron formed an alliance in 2009, but the alliance was unsuccessful.
The party fought under the unlikely name of UNCUNF in the 2010 General election but not one candidate was elected.
The arrangement also cost the Ulster Unionist Party its one MP. Lady Sylvia Hermon left the party in March 2010. On the other hand the former Ulster Unionist leader Lord David Trimble now takes the conservative whip in the House of Lords.
After the electoral setback the two parties distanced themselves from one another and the relationship turned sour. It has now been usurped by the confidence and supply arrangement which many NI Conservatives find uncomfortable.
The party faithful in Northern Ireland will want to hear what future plans the candidates have to grow the party as an electoral entity.
- ITV will stage the first head-to-head televised debate between the leadership candidates on 9 July, hosted by Julie Etchingham.