'I've been working as the party chief for the last 28 years. I want to see new leadership and I will provide all out cooperation to them,' Hasina was quoted as saying by a councillor.
But councillors did not agree with Hasina's proposal and requested her not to quit the party's helm.
Hasina was first elected president of the party in 1980 while she was living abroad. She returned home in 1981 and has been leading the Awami League since then.
Her political graph matches that of her arch rival and current opposition leader Khaleda Zia, who took over the Bangladesh Nationalist Party (BNP) in 1981 after her husband and then president Ziaur Rahman was killed in a military putsch.
The two have dominated the Bangladesh political scene since 1981.
In a related development, H.M. Ershad, the country's longest-serving ruler (1982-90), declared himself president for lifetime of his Jatiya Party Friday.
'From today there is no need for an acting chairman and I want to be the chairman of the party until death,' said Ershad, who presided over the council of his party.