Hundreds of babies have died or been injured while under the care of Nottingham University Hospitals (NUH) NHS Trust, which runs maternity departments at City Hospital and the Queen's Medical Centre.
The units, which are currently rated by the health watchdog as requires improvement, remain the focus of the largest inquiry of its kind with about 2,500 cases being examined.
In June, Streeting announced a national investigation into maternity care in England, with victims of scandals such as Nottingham set to be involved.
Ms Ockenden said she had discussed the issue with bereaved families in Nottingham and said she "felt confident" there was a way that she could take on more without impacting the work she has done so far.
She said she would be willing to chair the inquiry in Leeds and a maternity review at University Hospitals Sussex NHS Trust, describing it as an "honour" to be considered.

