Jagdish Singh said it was "troubling" to find out that tallow - derived from beef or mutton, but sometimes pork - was used to manufacture the fiver.
He joined a number of Hindus in urging the notes be banned from temples, where meat products are forbidden.
A petition to ban the new £5 notes has attracted more than 100,000 signatures.
It calls on the Bank of England to "cease to use animal products in the production of currency that we have to use".
Hosted on the Change.org website, it states that tallow is "unacceptable to millions of vegans, vegetarians, Hindus, Sikhs, Jains and others in the UK".
The Bank of England began issuing the polymer notes in September pledging they "last longer, stay cleaner and are harder to counterfeit than paper notes".
Its only response so far to the petition has been in a statement to "confirm that the polymer pellet from which the base substrate is made contains a trace of a substance known as tallow".
Critics say there are plant-based substitutes that could be used in its place.
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.
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.
Speaking to the BBC's Asian Network, Coventry-based Mr Singh said: "Every time I come across a £5 note I'll be reminded that it contains meat by-product."
He said no animal by-product should breach the sanctity of a gurdwara [Sikh temple].
Gauri Das, managing director at the Bhaktivedanta Manor Hindu temple in Watford, is calling for the notes to be banned from his site immediately.
"Our temples prosper on the charity of all of our members," he said.
"Our ethos is not to harm animals. It's problematic for us because we're implicated in the process. So it's immediately become a matter of concern for our community."
Meanwhile, the president of one of the largest Hindu temples in Leicester has urged worshippers not to give charitable donations with the new £5.
Vibhooti Acharya, from the Shree Sanatan Mandir temple, said it was a "matter of choice" but it would be putting up notices to make the community aware of the situation.
She added: "There needs to be a decision made between committee as to whether we accept £5 notes in religious ceremonies in future."
The revelation the UK's new plastic £5 note contains a small amount of animal fat is "extremely offensive", a Sikh activist has said.