The commanding centre-half has been strongly linked with a return to former employers Nottingham Forest - with the Reds reported to have failed with three bids during the summer window.
Forest have turned their attention to Morgan as he has entered the final year of his current deal with Leicester.
The 29-year-old is, however, prepared to prolong his stay at the King Power Stadium and bring an end to the rumours which are threatening to run into the New Year.
Morgan told the Leicester Mercury: "I'm happy. The manager has shown me nothing but love. It would be nice to be offered a contract.
"There was a lot of speculation. But I am just here to focus on my job and do what I do best, which is play football for this club. I can just concentrate on my football.
"The window is closed now so there will be no more speculation."
Nigel Pearson has always stressed that Morgan remains a key part of his plans and he has reiterated that he would like to see fresh terms agreed.
He said: "Our skipper is very, very important to us so hopefully we can do something there."