In an attempt to clarify the PM’s position, a No 10 spokesperson added: “He also stressed that it is an ongoing independent process. It is essential that it is allowed to run its course and the outcome respected.
“Wherever wrongdoing is established, those responsible should face the appropriate consequences.”
Man City deny any wrongdoing and have until Friday to submit an appeal.
Sources told BBC Sport that City were expected to claim that key sponsorship deals were funded by the Abu Dhabi government, rather than the club’s owners, as part of an appeal.
Among the independent panel’s findings were that City arranged “sham contracts” with a number of commercial partners as part of a scheme to disguise secret funding of more than £830m.
The Premier League says City “systematically broke Premier League Rules for nearly a decade”.
In an interview on Wednesday with the BBC’s political editor Chris Mason, Burnham – who was mayor of Greater Manchester until June – said he was “not getting directly involved in the process” and that it must be undertaken independently by the Premier League.
He said the situation was “contested by Manchester City” and it would be “wrong” to “jump to a conclusion”.
Asked if he was worried the club’s owners since 2008 – led by Sheikh Mansour, a royal and the United Arab Emirates deputy prime minister – might sell up, Burnham replied: “I would be really concerned to lose them.
“They’ve been such a huge partner in the building of modern Manchester and obviously the building of Manchester City into the global force that it is.
“I can’t intervene more in the process, it would be wrong for me to do that.”
The Tories have accused Burnham of “publicly backing the owners even as the investigation continues” and questioned whether “political pressure” was being applied.
Shadow minister for sport Louie French called for the government to “keep politics out of football and ensure the Premier League’s independent commission can do its job free of outside influence”.
“The club face unprecedented allegations and fans are horrified by what they are reading in the papers,” the Tory MP said.
“Yet Burnham is publicly backing the owners even as the investigation continues. Fans will rightly ask whether political pressure is being applied here and how it relates to the owners’ well-reported investments in Manchester.
“Football must be governed independently, with the rules applying equally to everyone. Fans deserve confidence that no club or owner is above those rules.”
And Lib Dem MP Anna Sabine, the party’s spokesperson on sport, said: “Fans of every club deserve to know the rules apply to everyone… Burnham needs to remember he’s prime minister for everyone, not just Manchester.”
The chairwoman of the Commons Treasury committee, Labour MP Dame Meg Hillier, wrote to HMRC on Thursday to ask for reassurance that it had “seized of the importance” of the case for any potential tax implications.