Shaikh Sabah, who has run Kuwait's day-to-day affairs for years, will continue as de facto leader under Shaikh Saad Al Abdullah Al Sabah, who was named amir on Sunday.
Aljazeera's correspondent in Kuwait reported that key figures in the ruling family met on Friday morning at the house of the prime minister and asked him to shoulder his constitutional responsibilities as the amir was largely incapacitated by illness.
The elders urged Shaikh Sabah to initiate constitutional procedures to assume leadership of the country.
State-run television on Friday reported that ruling family members visited Shaikh Sabah "and called on his highness to accept his responsibilities in continuing to lead the blessed procession of good and to reach the shore of safety".
Footage showed family members kissing Shaikh Sabah in congratulations.
The carefully worded announcement seemed to be an effort to prove to Kuwaitis that a leader was in place following the Sunday death of Shaikh Jaber Al Ahmed Al Sabah, who ruled the country for 27 years.
Shaikh Saad took over as amir
following Shaikh Jaber's death
As called for under the constitution, Shaikh Saad, the crown prince, was immediately named the new amir. But Shaikh Saad is ailing and it was not known if his health problems physically prevent him from taking the oath of office.
Members of the ruling family and parliament have held numerous meetings since Shaikh Jaber's death to discuss leadership of the country in light of the new amir's poor health.
Succession is a sensitive issue in the oil-rich emirate, which is accustomed to untroubled handovers of power.
Shaikh Saad is known for his closeness to the people. He is the eldest son of Kuwait's most popular amir, Shaikh Abdullah Al Salem Al Sabah, dubbed the "father of democracy" and founder of modern Kuwait.
Shaikh Sabah, the prime minister and half brother of the late amir, was seen as a strong candidate for becoming the next crown prince.