conto estero or conto per non

The foreign bank account ( conto estero ) is a current bank account, where payments and banking operations are carried out in a currency other than the official Italian currency, ie the euro - Normally Italian banks grant this type of account to people or companies who regularly carry out commercial transactions with foreign countries, whether or not they are resident in Italy. The account is checked by the Italian Exchange Office, and has use quotas that can be updated according to the owner's needs. The holding cost is normally high and can only be justified by a high number of annual transactions.The bank account for non-residents ( conto per non residenti ) , is a current bank account, in the official currency of the country in Italy, ie the euro - opened in the name of a person who does not have residence in Italy - every transfer inbound, generates an automatic exchange in euro with costs to be borne by the account holder. Outgoing transfers are in euros - If an outbound transfer is requested, in a currency other than the euro, the exchange will be made by the Italian bank, with costs to be borne by the account holder and exchange rates in currencies other than the euro, in line with the official Italian one. The cost of the operations is higher than that of the accounts for residents.Ugo , by Lifeinitaly.it

Unicredit do a conto nonresidente, which costs only a few euros more than a resident account and gives you nearly all the functionality of a resident account. 

Fineco even gives it for free - however it is good to remember that the account for non-residents does not allow the use of checks, but only ATMs and rechargeable credit cards

Fineco (Italian) also has a base in the UK, this is their site link. Not too sure why you need it, but if it is just for bills it may be worth looking at a combination of a pre-paid Euro credit card and/or something like Transferwise or Currencyfair.

Sorry Steve , to opena an ITALIAN bank accont , the requesto to be send to the ITALIAN BANK ORGANIZATION - IN ITALY - do not exist possibility to open an Italian bank account based in Itay , via Italian bank office , placed OUTSIDE the  Italian workin organization - this cause PRIVACY LAWRegarding assegno - The italian Banca d'Italia policy , and Italian Financial laws . Require an Italian Address ( residential address ) , necessary to send comunication about non PAYED assegno ( Assegno PROTESTATO ! )For this reason to the not residents bank accounts , is denied to have assegni bancari -Bank Check

The company is the same (Unicredit/Fineco), but the link is for the UK and can be opened on-line, it's basically a UK Euro account. The OP would need to look at it to see if it is suitable for what they want to do.

Steve , yes , are the same - but not possible open a bank account based in Italy - only in uk based  -the remember in UK DO NOT HAVE EURO CURRENCY AS NATIONAL CURRENCY an EURO BANK ACCOUNT in UK = CONTO ESTERO ( not in national currency - i believe even in UK , this type of bank account are ONLY for FIRM that deal in international currency  business )

Unicredit do give nonresident cheque books to holders of nonresident accounts.

it is a real novelty, I do not understand how it can do this, it is contrary to the current legislation, maybe the bank agency asks for a customer domiciliation?

thanks, I will open it in Italy. I will ask unicredit.

conto corrente unicredit non residenti - fixed  expenses http://www.conto-corrente.it/il-conto-corrente-per-i-non-residenti/Pdf - file >  https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=2ahUKEwiyuam1_b_hAhUhMewKHYrOCDIQFjAEegQIBBAC&url=http%3A%2F%2Fambasada.it%2Findex.php%3Fapp%3Dcore%26module%3Dattach%26section%3Dattach%26attach_id%3D11499&usg=AOvVaw374GkPR4BOtjkDAx_p1-vm