My suggestion would be to use a service like TransferWise to transfer your money to a relative / someone you trust’s UK bank account & then transfer your money to Monzo from there.
Otherwise you’ll be charged a conversion fee by your bank when you transfer the money from your Euro bank account to Monzo’s account & since it’s a bank, the fee probably won’t be very forgiving.
It’s also worth bearing in mind that you can’t transfer money from your Monzo account to a bank account (yet - not until the current accounts launch) so you’d need to withdraw it via an ATM or possibly, using PayPal in order to move it into your newly opened UK bank account.