Efforts to restore power-sharing at Stormont and a strategy to deal with the impact of Brexit are expected to dominate the discussions.
The start of parliament has been delayed since last Thursday's election, a gamble May took to strengthen her hand in talks to leave the European Union but which has left her scrambling for a deal to keep her in power.
But Stormont parties have a June 29 deadline to end the impasse and reach consensus on re-establishing a devolved administration in the region.
"The circumstances have changed dramatically with the election", he said.
"Those two things coming together hopefully mean there is a real confluence and perhaps that means we'll have the impetus we need to bring back power-sharing". He also highlighted the role of Scottish Tories, led by Ruth Davidson, who are opposed to a hard Brexit.
Arlene Foster said it was "right and proper" that her Democratic Unionist Party would vote for May's Queen's Speech next week.
Selfie - le Cose Cambiano: Anticipazioni Puntata del 16 giugno
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Mr Cameron said Parliament "deserves a say" and urged the Government "to consult more widely with the other parties" on how best to achieve a deal.
"We will go in to speak with Sinn Fein again on Monday morning to try and get that set up as quickly as possible, because devolution works and works for everybody in Northern Ireland".
More recently he has overseen Ireland's welfare system.
"However, while talks are ongoing it is important the Government gets on with its business and we are confident there will be sufficient support across the House for passing the Queen's Speech".
The announcement by Andrea Leadsom suggests that Prime Minister Theresa May and the Northern Ireland-based Democratic Unionist Party have struck an agreement or are close to one.
The 1998 Good Friday Agreement - also referred to as the Belfast Agreement - commits the United Kingdom and Irish Governments to demonstrate "rigorous impartiality" in their dealings with the different political traditions in Northern Ireland.