He succeeded Walter Onnoghen, ousted for allegedly failing to declare some assets traced to him.
Mr Kalu in a statement said his decision to step down is in the interest of the ruling party and the country at large.
Mr Buhari made the presentation of the appropriation bill to the joint session of the lawmaking arm of government on Tuesday.
Mr Oshiomhole, a former governor of Edo State, in April, said his party will not cede any position to members of the opposition particularly those from the Peoples Democratic Party (PDP).
If it becomes law, principal officers of the legislative arm of government at the federal and state levels are going to be the beneficiaries.
The report said the role played by the former Lagos State governor is aimed at positioning him for the presidential contest in 2023.
This was made known on Tuesday by the Chairman, Senate ad hoc committee on Media and Public Affairs, Senator Adedayo Adeyeye while addressing journalist after plenary.
A photograph showing the senator-elect with a certificate has been trending online yesterday but the former governor has described it as fake.
He is representing Yorro state constituency.
Although Mr Abbo confirmed the incident actually took place, he said he was in the shop to ensure justice is done to his sister who was assaulted in the building.