Such a strong result for the bank would represent a surprise success, as markets have suddenly turned cautious on initial public offerings in recent weeks.
Lloyds is selling a 25 per cent stake in the new high street bank, floating it on the stock market.
It had initially given a price range of 220p to 290p, and it is understood that the shares are now likely to sell in the top half of that range.
As a result, the bank can expect to be valued at around £1.3bn to £1.4bn, or 15 per cent short of its book value of £1.6bn.
The price range has not formally been tightened, but it is understood the investment banks marketing the shares to investors have verbally informed them of the high demand for the stock.
Analyst Gary Greenwood from Shore Capital yesterday argued that even at 290p the shares could be undervalued.
“We believe the shares have been priced to go and represent an enticing investment opportunity, even at the top-end of the proposed valuation range,” he said.
Despite the bank planning to wait until 2017 before paying a dividend, overseas investors are thought to be attracted to the quality of TSB’s loan book and its value as a household name.