Lopez and Young have player options on their contracts, and if they choose to opt out then they can become free agents.
However, Brooklyn own the 'Bird Rights' on both players, and so they will be able to offer them more than any other team in the NBA.
"Internally, the next big step is to keep Brook and Thaddeus here," King said at his annual end-of-season news conference.
"They both have shown indications they want to be here in their exit interviews, and we want them here, so it's incumbent on us to get that done."
The Nets were eliminated from the NBA playoffs earlier this week, losing to the Atlanta Hawks in the first round.