Young held a $10.2 million player option, but he was expected to opt out and test the market.
The Nets, who want to retain him, hold Young's Bird rights, which could enable them to pay him more than any other team could.
Young, who turned 27 on Sunday, averaged 13.8 points per game and shot 49.5 percent from the field after being acquired by the Nets at the trade deadline from the Minnesota Timberwolves in exchange for Kevin Garnett.
Nets center Brook Lopez is also expected to opt out of the final year of his contract and become an unrestricted free agent. He has a June 29 deadline to inform the Nets on whether he is accepting his $16.7 million player option for 2015-16.
Asked if Lopez is a max player, King responded, "Next question." King responded the same way when asked if Young is a max player.
Lopez, 27, who has a history of foot injuries, averaged 17.2 points and 7.4 rebounds per game last season. After a slow start, he came on at the end of the season to carry the Nets into the playoffs.
The Nets have repeatedly tried to trade Lopez in the past, but King feels "pretty confident" that they will be able to keep their starting front line together.
"We have some tools in our favor," King said, alluding to, among other things, the Bird rights the Nets have on both Lopez and Young.
The Nets have also extended forward Mirza Teletovic a $4.2 million qualifying offer, making him a restricted free agent, King said. Teletovic, a floor-spacing big man, missed significant time due to blood clots in his lungs in 2014-15.
King is not confident that the Nets, who hold the 29th and 41st picks, can move up to secure a higher pick in Thursday's NBA draft.
"Do I think we can move up? I don't think so, but you still have to go through the process and find out," he said. "The price to move up is high, but it tends to go down when you get closer to the draft."