The Suns are waiting on a formal response to a contract extension tabled to King as several clubs in Melbourne circle the AFL's leading goalkicker.
And Gold Coast are understood to have increased their offer to emerging attacker Walter, who is also a target of cashed-up rivals.
Victorian-born King will be a restricted free agent at the end of the current season, his eighth at the Suns.
"Free agency is part of our game now," Suns coach Damien Hardwick told reporters on Wednesday.
"It's his opportunity to explore; we understand that and respect that.
"What we are incredibly grateful for is the eight years he has provided us with. And I want him to spend another eight here."
Hardwick said Gold Coast won't pressure King to make a decision.
"There's no time frame on these things - he can decide this week, he can decide next week, he can decide in four weeks, he can decide at the end of the year," he said.
"What we do know at the moment is he's an important player and he's ours."
Walter has played only seven games this year, his third at Gold Coast, and falls off-contract at the end of the season.
The Perth-born 21-year-old is fielding million-dollar a season offers from numerous clubs including Essendon, Collingwood and Carlton.
"Great key forwards are hard to find and we're very fortunate we have a number of those guys - and we want them all to stay here," Hardwick said.
"We want that talent here and we've got to pay the price, this is the marketplace and state of the game we're in.
"Tasmania is coming in (to the AFL in 2028) so talent is at a premium.
"You have got to keep your very best players and we think Jed is one of them."
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