MPs have been debating John McDonnell's private member's bill - the Bank of England (Appointment of Governor) Bill - at second reading in the House of Commons on 6 July 2012.
Mr McDonnell's bill would require MPs to approve Bank of England governors.
The Labour MP told the BBC in May: "Under the government's new legislation, the governor of the Bank of England will gain a vast range of new powers, more than any governor in our history.
"Therefore there should be a decisive, democratic input into the appointment of the new governor.
"Making the appointment subject to parliamentary approval will ensure this significant decision is not just left in the hands of the Chancellor and the person appointed will demonstrably have the confidence of Parliament."
In May Chancellor George Osborne announced that the Bank of England would be given a key role in regulating the UK financial sector.
Mr McDonnell defied odds estimated at 58,000 to 1 to top the annual Private Member's Bill ballot for two years in a row.
MPs' names are selected at random, with 240 having entered the draw this year.
The bills are a way for MPs to propose legislation on a subject of their choice. Those near the top of the list get Commons debating time, although they only rarely make it to law.