Lampard spent 13 years at Chelsea as a player, and was twice appointed as the club's coach, including a short stint as caretaker manager in 2023.
The former England midfielder, who won three Premier League titles, four FA Cups and one Champions League title with Chelsea during his playing career, led the club to the FA Cup final as manager in 2020.
King said Lampard was in a "happy place" after winning his first trophy as a manager when Coventry secured the Championship title this week.
"You can see how emotionally connected (Lampard) is with the city, you can see what it meant to him to get promotion and then the title and I think he's found a happy place at the moment," King told the BBC in an interview published on Thursday.
"It doesn't mean there's not going to be some staggering bid for him in the summer and he'll have to make his choices, because he's shown his credentials as a high-quality head coach, but I'm not going to comment on that, I can't control it.
"I just think he's happy, I'm happy, everybody's happy, we've just got smiles on our faces and we don't worry about what may or may not happen for the head coach."
Lampard's contract with Coventry runs until June 2027.
Chelsea sacked manager Liam Rosenior on Wednesday amid a five-match Premier League losing streak in which the eighth-placed club failed to score a single goal.
Manager Andoni Iraola, who will leave Bournemouth in the summer, has also been linked with Chelsea in British media reports.
The London-based club will head to Wembley on Sunday to take on Leeds United in the FA Cup semi-finals.