Shares in Lloyds, owner of Bank of Scotland, soared 5.5% yesterday, after it reported a 21.7% rise in first quarter underlying profit to £1.8 billion.
The group benefited from another £400 million dividend from Scottish Widows, taking the total remitted by its Edinburgh-based insurance arm to £5.5bn since 2009.
Lloyds chief executive Antonio Horta-Osorio said the bank would seek to sell a minimum of 25% of TSB when it goes to the market.
Finance director George Culmer said: "We would be hopeful there will be an announcement made by the end of June subject to getting the financial approvals."
He added: "There will be a retail element."
The flotation comes after the collapse of a deal to sell the portfolio, which includes 185 former Lloyds TSB Scotland branches, to Co-operative Bank. Co-op bank later revealed it had a large capital shortfall and a restructuring left it in the hands of hedge funds.
Lloyds must sell the business as a condition of the £25bn bail-out of the bank following its rescue takeover of Edinburgh-headquartered Halifax Bank of Scotland during the financial crisis.
TSB, which was launched as a separate strand by Lloyds in September, is expected to be valued at around £1.5bn.
TSB cost Lloyds £172m in the quarter, including building it and dual running costs. Separating the bank out has cost Lloyds £1.6bn so far.
Having originally had a 43% stake in Lloyds, the Government retains a 24.9% holding after raising £7.4bn from selling shares to investors in two tranches since last summer.
Lloyds's profit boost came from falling costs which dropped to £2.3bn in the quarter, down from £2.5bn for the last three months of 2013.
Mr Horta-Osorio's plan for the bank set out in 2011 envisioned cutting 15,000 roles.
Mr Culmer said yesterday that Lloyds currently has no plans for further redundancies.
"We are supporting and benefiting from the UK economic recovery and are delivering better underlying profitability as well as improved returns for shareholders, from a stronger, lower risk balance sheet," Mr Horta-Osorio said.
Analysts were surprised by weaker than expected revenues of £4.5bn for the three months to the end of March, down 7% on the same period last year. This was due to lower than anticipated loan growth.
However, the weakness in income was offset by a sharp improvement in loan impairments at £431m, down 57% on the same period last year.
This prompted investors to reappraise Lloyds's shares and they closed the day up 4.14p at 79.5p.
Chris Beauchamp, market analyst at IG, said: "The results today are likely to boost appetite for the TSB IPO (initial public offering) among investors."
Lloyds could pay a dividend as early as May next year.
Before the financial crisis Lloyds was among Britain's highest dividend paying stocks.
Some analysts had expected Lloyds to pay a dividend for 2013, but this was ruled out after the bank warned in February it would need to take a further £1.8bn charge to compensate customers who had been mis-sold payment protection insurance.
The money transferred to Lloyds from highly profitable Scottish Widows has been very useful to the bank as it has sought to fill in a capital shortfall identified as £7bn by regulators last summer.
It is understood that the £400m paid over during the quarter was unrelated to the sale of Scottish Widows Investment Partnership but came from profits from ongoing business.
Lloyds improved its guidance on future margin and impairment charges and said it expected a 2014 net interest margin of 2.4%, an increase of around 10 basis points on previous guidance.
The margin is a major driver of profits for Lloyds and improved to 2.32% in the first quarter, up from 1.96% a year ago, which drove a 10% rise in net interest income.