The Bank said the problem "had only just come to light" and it was treating the concerns with "utmost seriousness".
Vegans have expressed anger because the new polymer fiver contains a small amount of tallow, which is derived from animal waste products.
A petition to ban the note has attracted more than 100,000 signatures.
The tallow is used in an early stage of the production process and is "an extremely small amount", the Bank said.
"We are aware of some people's concerns about traces of tallow in our new £5 note. We respect those concerns and are treating them with the utmost seriousness," its statement said.
"This issue has only just come to light, and the Bank did not know about it when the contract was signed.
"[Supplier] Innovia is now working intensively with its supply chain and will keep the Bank informed on progress towards potential solutions," it added.
The petition, hosted on the Change.org website, calls on the Bank of England to "cease to use animal products in the production of currency that we have to use".
It states that tallow is "unacceptable to millions of vegans, vegetarians, Hindus, Sikhs, Jains and others in the UK".
A number of Sikh and Hindus have also urged the notes be banned from temples, where meat products are forbidden.
Hindus believe cows are holy and sacred, and many do not wear shoes or carry bags made from the skin of cattle that has been slaughtered. Practising Sikhs are strict vegetarians.
The response from the UK's Hindu and Sikh communities began to gather pace after vegans and vegetarians voiced their feelings on social media on Tuesday.
The new plastic £5 note was introduced in September and is more durable than the previous one. It is expected to last an average of five years - compared to two years previously.
The Bank of England says its supplier is working on "potential solutions" to the issue of animal fat in its new £5 notes.