For a Labour prime minister there are yawning policy gaps on every issue from climate and immigration to Iran and Nato.
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One intriguing blueprint is offered by another mayor: Zohran Mamdani of New York.
A democratic socialist, he is ideologically opposed to Trump yet has got on well with the president, who appears to respect him as a charismatic populist – and a winner.
Philippe Dickinson, deputy director of the transatlantic security initiative at the Atlantic Council, said: “Mamdani is obviously on a very different side of the political spectrum but his identity isn't 'I'm the anti-Trump guy'.
It's about his policy platform domestically in New York. For Andy Burnham, there are potentially some lessons there.”
Among them, Dickinson said, are the charm offensives by Mamdani and Mark Rutte, the secretary general of Nato.
“It's politicians who convey an aura of confidence and ease in their own skin and can present themselves as people who are eye-to-eye with Trump on certain things.”
One clear commonality between Burnham and Trump is their shared view that mainstream politics has not worked for ordinary people.
Trump may admire Burnham's desire to shake up the staid British political establishment.
Sawetz said Trump's approach had long been to “establish the power dynamic quickly, whether through public criticism, personal remarks or by goading new counterparts into a very public response.”
“Burnham won't take that invitation,” she said.
“He may brush off the occasional quip, but we know he doesn't favour open political confrontation or the kind of prolonged exchanges we've seen with Sadiq Khan in London.”
