'statement saying that' is correct and usable in written English. You can use it when you are referring to an expression or statement that is being attributed to someone or something. For example: The prime minister released a statement saying that the government would be implementing a series of tax reforms.
Instead, Labor backbencher Graham Perrett simply issued a statement saying that Brough needed to explain himself.
Young initially issued a statement saying that use of his song was "not authorized".
Arch issued a statement saying that company executives "respectfully disagree" with the agency's decision.
Within hours, the Chechen leader issued a statement saying that "Kadyrov's Path" had been closed.
The British consulate in Miami issued a statement saying that next of kin had been notified.
Collar closed the statement saying that Yours Truly "makes the most of her talent".
They were more inclined to agree with a statement saying that overweight will disappear by itself as children grow.
Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.
Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com