Fifty-four Tory MPs must submit letters of no confidence to Sir Graham Brady, the 1922 committee chairman, to trigger a ballot. Mr Harper emailed his letter of no confidence last week.
Mr Harper added: “I think my colleagues will have enough evidence to make a decision, and I very much hope that a majority of them reach the same conclusion that I have. We can resolve this matter then, and later we can give the country the fresh leadership it deserves.”
Asked if this would happen “before the summer break”, he replied: “Yes.”
Mr Harper, who said he would not stand again for leader after running in 2019, declined to name a successor to Mr Johnson.
“I don’t want this whole debate this week to be about me,” he said. “It’s about whether he’s the right person to lead us.
“Look, we’ve got fantastically talented people in the Conservative Party, and the way this works is if there’s a vacancy, if we vote against him in a vote of no confidence and there’s a vacancy…
“Good men and women will put their names forward and there’ll be a contest and they’ll be tested in the contest, and we will pick the right person to lead the party and become Prime Minister.”
Listen to Christopher Hope’s interview with Mark Harper on Chopper’s Politics, The Telegraph’s weekly political podcast, using the audio player at the top of this article or on Apple Podcasts, Spotify or your favourite podcast app.