Senior club sources have said they found the comments "absolutely bizarre", and that it was "at best naive and at worst quite manipulative, and that is not a good look".
A different source said it was "clumsy", while another said they were surprised that the prime minister had "misspoken".
The entire process involving the charges against Manchester City is independent of the Premier League.
An independent commission held a 42-day hearing in 2024 which led to the verdict announced last month, with a sanctions hearing still to come.
A separate independent appeal board will hear City's appeal. The club continues to deny any wrongdoing.
However, when asked about potential interference in the process off the back of Burnham's comments, one senior club source said: "Everyone would have a degree of concern. How much would depend on how much you read into that comment.
"I think everyone is concerned. Because no-one is stupid around the table. We understand the geopolitics as well. We understand the investment they have made in our country. I don't think anyone could doubt that or criticise that. But this is different, this is sport, this is our national game."
Another club source, however, said the degree of scrutiny on Burnham now means they think any interference would be impossible.
Burnham's comments came in an interview with the BBC's political editor Chris Mason the next day.
When asked about 'Brand Manchester' taking a hit, the Prime Minister drew a parallel with a previous Premier League case with Everton, saying he still held the view "firmly" that they "weren't fairly treated".
Everton were deducted 10 points, later reduced to six on appeal, by an independent commission for breaching Premier League financial rules.
Asked if he was concerned that Manchester City's owners would now sell the club, Burnham said: "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 do actually thank the City Football Group and the wider ownership for the money that they didn't just put into the Etihad [Stadium] and the campus around it but also into the city," he added.
Burnham's comments provoked strong criticism.
The following day, a No 10 spokesperson sought to clarify the comments, saying: "As the prime minister made clear, the initial judgement is serious and there can't be any suggestion that anyone is above the rules. 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.
"If wrongdoing is established, those responsible should face the appropriate consequences."
However, this statement was amended following further criticism over the use of the word "if" - which was changed to "wherever".
The Premier League is viewed by governments of different political stripes as one of the country's major cultural exports, so ministers are eager for the row not to cause lasting damage to its brand.
Additionally, being a football fan is a core part of Burnham's political persona.
He is an Everton season ticket holder and, prior to becoming prime minister, he spoke of hoping to continue sitting in his usual seat at Hill Dickinson Stadium - but this has not been possible because of security concerns, as was the case for Sir Keir Starmer, his Arsenal-supporting predecessor.
Last month Burnham floated the prospect of lifting the ban on football fans drinking in the stands, though it was quickly clarified that this is not in fact government policy.
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