Following the hearings, the MEPs on the Committee on Economic and Monetary Affairs will give their recommendations, which will be followed by a secret ballot in November.
The candidate will succeed Daniele Nouy as head of the Single Supervisory Mechanism (SSM), where they would oversee 119 of the currency union’s biggest lenders.
The winner will take on supervising the part of the bank struggling with the results of the last economic crisis, including unpaid loans.
Ms Donnery is seen as a favourite for the top job after being backed by the president of the European Central Bank, Mario Draghi.
However, she is facing challenges from MEPs from southern Europe who are concerned that the Irish central banker would be too harsh on the region’s banks.
She has worked to force banks to get rid of their balance sheets of non-performing loans and was behind SSM proposals last year which legislators criticised and changed.
The winner of the chief job will be in charge of ensuring lenders shrink their bad credit and prepare for an influx of big investment banks when the UK leaves the European Union.
Chair of the Autorite des Marches Financiers, France’s markets regulator, and Andrea Enria, head of Italy’s European Banking Authority are also going for the top job.
An EU diplomat said: “It will be a close call”.
Italian professor, Antonella Sciarrone Alibrandi, said: “Donnery would be seen positively by those who want faster progress on non-performing loans.
“But we need someone who can embody the European interest and avoid further division by not being too aggressive.”
If Ms Donnery succeeded in getting the role, it could make it harder for the Irish Central Bank Governor Philip Lane to be appointed to the ECB’s Executive Board next year as they are both from the same country.
Ms Nouy’s term began in 2014 and must end at the end of 2018.
The appointment will be finalised in December.