Mr Dunne, who was Jeremy Hunt's campaign manager in the 2019 leadership election, said Mr Hunt exhibited the qualities he would want to see in Britain's next prime minister.
"If the Prime Minister resigns, I'm very relieved," said Mr Dunne.
"He has recognised the reality of his position. The whole country wants to see this resolved as quickly as possible."
Mr Dunne said he hoped the 1922 Committee, which represents rank-and-file Conservative MPs, would ensure that a new leader could be elected before the summer recess.
He said it would be a matter for the Cabinet to decide whether Mr Johnson should remain in post while the leadership election is carried out, or whether an interim leader – possibly deputy PM Dominic Raab – should take over.
Speaking of Mr Hunt, Mr Dunne said: "If he stands, he would be the best placed candidate to provide the integrity and competence that we need to bring back into the role."
He said whoever took over would need to demonstrate they could bring back some integrity to the role.
"I think those who have been at the heart of the Government, certainly over the past 12 months, have got some serious questions to answer," said Mr Dunne.
He said whoever became leader would become Prime Minister overnight, so it would not be right to appoint a novice leader.
"They will need to have had experience of senior Government positions, competence and experience," he said.
Mr Dunne said another requirement would be that the new leader would need to have a coherent plan for the challenges the country face, with both the economy and the war in Ukraine being important factors.