They are both allowed. Personally, although I don't think it is a rule of grammar, my preference is:
If the verb is important to how the dialogue should be read, put the verb closest to the quote.
But if the verb is "said" (or "replied" or "answered" or "asked") and does not really inform how the dialogue should be read, then on the tail end I put the noun first and verb last, because the reverse sounds old-fashioned.
"Stop now!" shouted Jason.
Jason shouted, "Stop now!"
Jason whispered, "Get ready to run, kid."
"get ready to run, kid," whispered Jason.
But with "said", it is verb-noun on either end:
"Stop now!" Jason said.
Jason said, "Stop now!"
Jason said, "Get ready to run, kid."
"Get ready to run, kid," Jason said.
And as you note, with pronouns "said he" sounds old-fashioned. Start with the Noun-verb, pronoun-verb set:
Jason whispered, "I love you." sounds fine.
He whispered, "I love you." sounds fine,
"I love you," Jason whispered. sounds fine,
"I love you," He whispered. sounds fine.
Now the verb-noun, verb-pronoun set:
Whispered Jason, "I love you." sounds weird,
"I love you," whispered Jason. sounds fine,
Whispered he, "I love you." sounds weird,
"I love you," whispered he. sounds weird,
Those are all possible combinations, and the rule seems to be Noun-Verb or Pronoun-verb always sounds okay, before or after the dialogue, but the only time Verb-Noun sounds okay is after the dialog. Verb-Pronoun never sounds okay.
Like I said, I don't think this is addressed by rules of grammar, but aesthetically, it is how I write.