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 the outcome of the meeting between President Muhammadu Buhari and lawmakers-elect of the green chamber when the former hosted them to a dinner Tuesday evening.
Mr Lawan was the preferred candidate of the ruling party when the formation of the 8th Senate was due in 2015.
Except he backs down, Kalu might trigger what played out during the formation of the 8th Senate in 2015 to repeat itself.
Mr Kalu said the party either hand him the position or have him vie for Senate President.
The PDP lawmakers were reacting to the endorsement of Senator Ahmed Lawan by the APC wherein the ruling party picked him as its preferred candidate for the exalted office.
The YPP is the political party on which Mr Ubah ran for the senatorial race and was elected.
Mr Ndume, a former Senate Leader made his intention known in an intent letter he wrote to Adams Oshiomhole, the national chairman of the All Progressive Congress (APC).
This is not the first time Mr Lawan who first became a senator in 2007 is indicating interest for the exalted position at the red chamber.
The bill passed its second reading on Tuesday.