The pair tabled an £825 million offer for the lender in March at 330p a share but on Monday upped this to 340p.
Shawbrook has so far urged shareholders to rebuff the overtures, and investors now have until June 19 to decide whether to accept or reject the latest offer.
Lindsey McMurray of Pollen Street Capital and Cedric Dubourdieu of BC Partners said: “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.
“We continue to share the Shawbrook management team’s vision and look forward to working with them in the future to build on the company’s strong foundations as a specialist lender.”
Shawbrook, which specialises in lending to small and medium-sized businesses, warned earlier this year in an investor circular over “significant” costs of the bid, at around £4 million if it is aborted and up to £12.5 million if it succeeds.
The takeover approach comes after Pollen Street floated Shawbrook on the London Stock Exchange at 290p a share in April 2015, valuing the bank at £725 million.
Shawbrook employs 600 people in 10 offices across the UK.