Sir Alan Duncan, the Europe minister, said there were "quite a lot of consequences" of Mr Johnson's failure to display a "conventional diplomatic manner" during his two years at the Foreign Office.
He claimed that one country's foreign minister had suggested he was Mr Johnson's "pooper scooper" because he had cleared up his mess so many times.
Mr Johnson's two years as foreign secretary were marred by a series of gaffes, including wrongly claiming that a British citizen imprisoned in Iran had been training journalists in the country, describing Africa as "that country" and suggesting Libya had a bright future if it simply "cleared the dead bodies away".
Despite his record, he has emerged as the frontrunner in the race to succeed Theresa May as prime minister.
Sir Alan admitted that his former boss was ahead in the contest and called on a third of the current 12 candidates to quit, saying it was "embarrassing" that so many had put themselves forward.
He suggested that Mr Johnson did not have "a conventional diplomatic manner", a problem which led to "quite a lot of consequences", and said his support for Brexit had angered other countries.
He added: "There was an amusing moment when one foreign minister said to me, 'Alan, what is it like being Boris Johnson's pooper scooper?'. Clearing up after him was quite a full-time activity."
Sir Alan admitted that Mr Johnson was "the man to beat" in the race but said there were "doubts about personal qualities".
He predicted that the former foreign secretary and all the other leadership candidates would have to accept in the future that Ms May had been right about Brexit.
The large number of candidates vying to succeed the current prime minister has prompted calls from some senior Tories for party rules to be changed to narrow the field.
Sir Alan backed the idea, saying the current number of contenders was "a bit embarrassing" and "a bit silly" and calling on contenders to withdraw if they fail to secure the support of at least ten Tory MPs.
He named Mr Johnson, Jeremy Hunt and Michael Gove as being in the "top league" of candidates, ahead of the "next layer" of Sajid Javid, Dominic Raab and possibly Andrea Leadsom. He also praised "young thrusters" Matt Hancock and Rory Stewart, who he said had "great qualities, but probably for the next generation"
But he added: "Everybody else should just admit to themselves that this is perhaps a cry for attention or a cry for 'excuse me can I be a minister when this is all over?' and I think they ought to have a bit more self-realisation and stop now."
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