The offer of 340 pence a share values Shawbrook at about 868 million pounds, up from the previous 842 million pound bid Marlin Bidco, the buyout vehicle set up by BC Partners and Pollen Street Partners, said in a statement.

The offer, which is a 27 percent premium to Shawbrook's closing share price on March 2, when the lender first received a bid from the private equity firms, would now remain open until June 19.

"After carefully considering market feedback we are pleased to be able to make an improved best and final offer, which we consider offers shareholders an attractive premium and compelling value" Lindsey McMurray of Pollen Street Capital and Cédric Dubourdieu of BC Partners said in a statement.

The private equity groups already hold 38.8 percent of Shawbrook shares and have so far received acceptances from investors holding another 6.6 percent of the stock, leaving them just under 5 percent short of the required 50 percent backing needed for the deal to go through.

The consortium first made its bid for Shawbrook in January offering 307 pence per share, upping it to 330 pence in March.

However so far Shawbrook's directors have advised shareholders to reject the offer.

Britain's so-called challenger banks have been increasingly seen as ripe for takeovers in recent months, bankers who advise on mergers and acquisitions have said, as a prolonged period of low interest rates has squeezed earnings and the pound's fall has made them cheaper for foreign buyers.

In June last year Shawbrook booked an additional impairment charge due to some irregularities in its asset finance business, sending its share price to a record low.

Shares in Shawbrook, which have already priced in an improved offer, were down 0.3 pct on Monday at 339 pence at 0706 GMT.

(Reporting by Noor Zainab Hussain in Bengaluru; Editing by Rachel Armstrong)

By Noor Zainab Hussain