The Co-operative Bank will announce on Monday that it has reached a deal with regulators to plug its £1.5bn capital shortfall, kickstarting a crucial week for the banking sector amid expectations that the government is preparing to signal the selloff of shares in the bailed-out banks.

The chancellor, George Osborne, is expected to use his Mansion House speech on Wednesday to indicate that he intends to privatise Lloyds Banking Group and Royal Bank of Scotland, whose chief executive, Stephen Hester, was forced out last week to accelerate a sell-off of the bank, which is 81% owned by the taxpayer. This weekend, Hester was reported to have said it could take 10 years to privatise RBS.

Osborne also intends to use the set-piece speech to a City audience to respond to the parliamentary commission on banking standards report, which, while potentially affecting the structure of RBS, will focus on measures to clean up the City in the wake of the Libor-rigging scandal.

The chancellor will be relieved that an agreement between the Co-op and the Prudential Regulation Authority (PRA) has been reached after months of speculation about the ambitions of the bank following a six-notch downgrade of its credit rating by Moody's.

Co-op is regarded as a key challenger to the big four banks at a time when the government is keen to bolster competition on the high street but had to pull out of the acquisition of 632 branches from the bailed-out Lloyds. Co-op Bank bondholders, many of whom are private investors, are expected to take a hit of as much as 30% as part of the agreement with the PRA to fill the gap, which has already forced the bank to stop corporate lending.

The rest of the capital may be found by selling off troubled property loans – largely inherited through a tie-up with Britannia building society in 2009 – and the planned disposal of its insurance businesses. It is expected that the banking arm will be given some additional time to bolster its capital under its recently appointed new management team, led by former HSBC banker Niall Booker.

There had been speculation that some parts of the wider Co-operative Group, which includes grocers, funeral homes and pharmacy chains, might need to be spun off to help prop up the banking arm, although it appears that such measures can be avoided under the deal with the PRA.

The future of both Lloyds and RBS is also expected to be debated this week by both the parliamentary commission on banking, chaired by conservative MP Andrew Tyrie, and the chancellor.

There are suggestions that the commission may call for a review of the structure of RBS by September but back away from any recommendation of a split into a "good" and "bad" bank.

The chancellor is thought unlikely to be specific in any timing and prices at which the stakes in the bailed-out banks will be sold off, although likely to believe that Lloyds is closer to being ready for privatisation than RBS.

Any sale of RBS is likely to require the bailed out bank to buy the dividend access share it issued to the Treasury when it joined the asset protection scheme that insured its most toxic loans. The share stops RBS paying a dividend to other shareholders and the Policy Exchange thinktank estimated last week that this could cost RBS about £2bn to redeem.

Osborne is expected to have to initiate discussions about this special dividend blocking share before sanctioning any sell off which RBS has said could take place before the May 2015 election. Last week, Treasury ministers played this down.

The banking commission's report is expected to call for measures to make individual bankers more responsible when problems arise and is expected to be published before the Mansion House speech after six months of evidence from when it began its first public meetings in September. It is also expected to make recommendations about bolstering competition in the banking sector.