Sky Sources say Leicester have already exercised their one-year option on 30-year-old defender Morgan, who otherwise could have left on a free transfer this summer.
Nugent, meanwhile, was out of contract and was being linked with Swansea and West Ham but is now close to finalising a new contract to help Leicester make an impact on their return to the Premier League.
The 29-year-old former England international scored 22 goals in 51 appearances in all competitions last season, following his 16 the year before when he was also top scorer.
Manager Nigel Pearson, whose contract was also due to expire this summer, was this week rewarded for guiding Leicester back into the top flight with a new three-year deal.
Last season, City broke the 100-points mark as they left their promotion rivals floundering and Pearson was awarded the Championship manager of the year award.