Mouratoglou is one of the strongest advocates for on-court coaching, even though his player never uses it. He said Williams’s sense of self-reliance was a weapon he would not dare muzzle.

“I think that her No. 1 quality is to be the best competitor ever, and I don’t want her to think that suddenly she needs someone to be a better competitor; that would make her weaker,” he said. “So, for me, it doesn’t make sense for someone like her.”

Mouratoglou, who has worked with Williams since 2012, owns a tennis academy that bears his name in France and has been a television analyst for ESPN and Eurosport. He said tennis had many untapped areas that could be mined for entertainment purposes.

“Coaching is part of the game,” he said. “If you show it, it’s an incredible adventure for the people to see.”

Before the Australian Open began, Mouratoglou even released a mobile game, Tennis Manager, which offers users the chance to simulate recruiting, training and strategizing with tennis players. Its slogan: “Shape a Champion; Become a Legend.”

Bajin, who worked alongside Mouratoglou for years when he was Williams’s hitting partner, comes down on the opposite side of the coaching debate, saying the rule should not be changed “to make it more interesting” to watch tennis.

“I think it takes away the purity that tennis was,” Bajin said. “My father put me out there on the court in order to become more independent, to learn how to handle problems, to learn how to be by myself, so that nobody can help me — and I think that still parents do that for their kids. That’s why they want them to start this sport, because it’s something special, and I wish we would keep it that way.”

But Bajin admitted that he, too, had occasionally tried to send illicit advice from the stands.

“Sometimes you try to sneak something by,” he said. “I’m guilty of that.”