Clayton scored on his senior debut against Macclesfield on Wednesday, where he was reportedly watched by scouts from 17 Premier League clubs - leading to suggestions he may be the subject of an approach from a top-flight team at some point.
But Gradi believes that kind of talk will only make things more difficult for the starlet.
"It doesn't do anyone any good, it raises the expectation level," he told BBC Radio Stoke.
"There's a high expectation placed on these kids. For instance, there's a line in the newspaper linking Max with Tottenham. His parents are terrific and that's why he's handling it very well, but sometimes the parents can get carried away and they put expectation on their kids.
"With the great rewards that there are now for players who make it to the top, there's definitely more pressure than there used to be."