This was according to a statement by the Senior Special Assistant on Media and Publicity to the Vice President, Laolu Akande.
Osinbajo had earlier directed that the sacked Daura should handover to the most senior DSS personnel.
Meanwhile, the Inspector-General of Police (IGP), Ibrahim Idris has absolved his men of blame from Tuesday's blockage of the National Assembly (NASS) by security operatives, saying that the development came to him as a surprise.
A blockade of Nigeria's parliament on Tuesday by security forces that prevented lawmakers from entering for up to an hour was done without the presidency's knowledge, said the acting president, who sacked the head of the security agency.
Acting President Yemi Osinbajo has terminated the appointment of Department of State Services (DSS) DG Lawal Daura.More news: Gates tells Manafort trial he assisted in filing false tax returns
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Seiyefa is to act as the Director General of the Department of State Security.
Saraki was also expected to meet with Mahmoud Yakubu, chairman of the Independent National Electoral Commission (INEC).
It is not enough to take populist approach of easing Daura out of office.
"Acting President Yemi Osinbajo, SAN, has described the unauthorized takeover of the National Assembly complex earlier today as a gross violation of constitutional order, rule of law and all accepted notions of law and order".