With less than three days to go before nominations close, the Bank is urging more people to make suggestions, via the Bank's website.
Favourite ideas so far include the sculptor and artist Barbara Hepworth, the author Beatrix Potter and fashion designer Alexander McQueen.
Whoever wins will replace the economist Adam Smith on the current £20 note.
The Bank's chief cashier, Victoria Cleland, urged as many people as possible to choose a name.
"There's certainly still time, and it would be great to get some more nominations," she told the BBC.
"If you've got time, tell us why - that will help us with the decision-making."
The suggestion must be for someone who worked in the visual arts, including architecture, sculpture, fashion or film-making. It cannot be anyone who is still alive.
The winner will be based on the strength of their artistic contribution, not on the basis of the number of nominations.
Nominations on the Bank of England's website will close at midnight on Sunday.
As many as 18,000 people have suggested a new face to put on the new £20 note, the Bank of England has revealed.