The 340p per share bid, which was upped yesterday from 330p, “undervalues Shawbrook and its prospects”, according to the bank. Its shares are currently trading at around 338p.
The share price has remained high after it leapt from 301p following BC and Pollen Street's initial £842m firm offer at the end of March.
Yesterday was the last chance which BC and Pollen Street had to increase their offer, ahead of a 19 June acceptance deadline.
But Shawbrook's board are intent on fielding off the consortium's unwanted advances, and said in a statement that the bank's "independent directors believe Shawbrook can continue to grow prudently over the medium term within its risk and return disciplines".
At the bank's annual general meeting (AGM) today, all of the board were re-elected.
Since the 10p per share increase was not enough to bring the challenger bank's board on side, the bidding consortium must wait to see if enough shareholders sell up to allow them to cross the 50 per cent minimum threshold.
At the last count, on Friday, the consortium owned around 46 per cent of the challenger bank, having received a few acceptances on top of the 38.9 per cent stake which Pollen Street already owned.
If BC and Pollen Street seize control of more than 75 per cent of Shawbrook, they will take the London-listed challenger bank private.
But if the consortium only ends up with between 50 and 75 per cent, it will remain listed.