Rosenior's fate was sealed by a fifth Premier League defeat in a row, leaving qualification for next season's Champions League almost impossible.
The Blues failed to score in each of those five league defeats which is their worst goalless losing league run since 1912.
With Rosenior sacked, several names have already been linked to the Chelsea job including former midfielder and club legend Lampard.
Lampard has just secured the Championship crown and Premier League promotion with Coventry and is a leading candidate to takeover the West London club.
A move to Chelsea seems unlikely
Chairman King is not convinced however and brushed such rumours aside this week now that their promotion is confirmed.
“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.
"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 has also dismissed rumours in the past, as he focused on winning silverware for Coventry which will only make Chelsea desire him even more.
"I want to win this league now after what we've done through the season because it will be a slight anti-climax feeling for myself if we didn't get the job done there, and that's a good pressure to have at this point, so I want to do that, and nothing else is taking me anywhere away than thinking about that."