“I’m probably about the only politician I know of who is actually willing to stand up and say that he’s pro-immigration,” said Boris Johnson. But that was then – 2013, the year in which he called for a one-off amnesty for illegal immigrants.

Johnson has travelled a long way since. The pro-Brexit camp’s best known face was accused of racism after he hit back at Barack Obama’s intervention in the EU debate by referring to “the part-Kenyan President’s ancestral dislike of the British empire”. Last night (Friday) he and his fellow Leavers were accused by Sir John Major of “morphing into Ukip” and playing the immigration card.

The former Mayor of London may soon be travelling to Downing Street. Although he characteristically dismissed as “cobblers” the idea that Cameron would step down after a Brexit vote next month, few Conservative MPs expect the Prime Minister to survive for long if that happens. Johnson would be the clear frontrunner to succeed him.

Immigration is not the only issue on which Johnson has changed his mind. I remember when he used to say the EU was good for the City of London. After his unexpected last-minute conversion to the Brexit cause, he insisted the City would “flourish mightily” outside the bloc. Johnson admits he has changed, but claims the EU has changed more.

Many Conservative MPs are convinced that Johnson made a calculated decision that, with two out of three Tory party members backing Brexit, backing a Leave vote would enhance his leadership prospects more than sticking with Cameron, as the Prime Minister expected him to do. The word on the Tory grapevine is that Sir Lynton Crosby, the Australian strategist who helped both Cameron and Mayor Boris win elections, advised Johnson to opt for Out. Johnson doesn’t deny it, telling The Spectator magazine: “Lynton is too good, he’s too nice, to be dragged into this.”

Johnson, who this week accused Cameron of predicting “World War Three” if we left the EU, is not having a good war so far. He was in typically bouncy, confident form when he launched his red battlebus in Cornwall. But Sarah Newton, a local Tory MP, dubbed it Boris’s “blunder bus” after he brandished a Cornish pasty without knowing it was EU laws that protect pasty-makers from being undercut by inferior imports.

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Like others in the Leave camp, Johnson has struggled to provide a vision of what life would be like outside the EU. First he talked about Britain being like Switzerland, then it was a Canada-style free-trade deal, then a special EU trade deal for the UK, now it is about “access” to the EU market – without having to allow EU citizens to come to Britain, of course.

Yet Johnson has also conceded that it might be hard to get such a wonderful deal from the EU if we left, telling Der Spiegel last August that there would be “several disadvantages”, including “some penalties”. Asked to elaborate, he told the Treasury Select Committee in March: “I’m sure there would be some [penalties]…I frankly have no idea as to what penalties they might be so foolish as to impose but, let’s face it, there would be some feeling of hurt perhaps on the part of some of our European friends and partners.”

Johnson’s two-and-a-half hour grilling by the committee was uncomfortable. Andrew Tyrie, its Tory chairman, was not impressed with what he called a “busking, humorous approach to a very serious question for the UK”. Tyrie found Johnson unable to substantiate his previous claims that the EU had banned children aged under eight from blowing up balloons and forbidden the recycling of tea bags. The word has spread at Westminster about Johnson’s performance. Even some fellow Outers admit it damaged his credibility. “We might look back on it as a turning point, when some people started to sell shares in Boris,” one said.

Johnson also had unfavourable reviews after appearing on the BBC’s Andrew Marr Show. Ruth Davidson, the Tories’ impressive leader in Scotland, tweeted: “Is it just me or is Boris floundering here? Not sure the bumble-bluster, kitten smirk, tangent-bombast routine is cutting through.”

No wonder George Osborne, the Chancellor and Johnson’s main rival for the Tory crown, speaks of the need for “sober, serious, principled” leadership. Some Tories talk up Michael Gove, Johnson’s fellow Vote Leave leader, as a more serious alternative to Boris. Gove is genuine when he says he does not want to be prime minister. But, to borrow Johnson’s words, if “the ball came loose from the scrum,” Gove might just run with it.

Johnson remains the hot favourite, even though some wise old Tory heads judge that he needs to prove himself in a big Cabinet post to show he is qualified for the top job – health, perhaps, where Jeremy Hunt is being tipped for respite care in another department if Cameron wins the referendum.