Bank accounts for non resident aliens in Spain

Not the question you asked, but are you aware of the Schengen Area 90 days in 180 rule?
"2 months in Madrid and 1 month in Barcelona" = 92 days, which exceeds you 90 day tourist allowance (assuming you are US citizens). So unless you cut a few days off you would need a visa.

To open a bank account you would normally need legally valid ID (legally valid by Spanish law, for a foreigner that would be a passport) and proof of address. But, if you are a tourist and only staying for < 90 days, they might not be interested.
Why do you want a bank account? For this period you can carry on using your own home account, and get money out of a cash machine using your home bank card.

Years ago when our son studied in Madrid for six months we tried for an account. He could not because he was a student and non-resident. Why would you need a bank account for 90 days or less? Given internet banking and ATMs it is easier to get by without a local bank account than it was 25 years ago.

Remember 90 days is not three month unless you are going in January. It was reported here earlier that a young lady thought it meant three months. The one day error cost her 500 Euro, missed plane, and a new plane ticket. So count carefully.

European banks rarely open accounts for USA citizens these days because of your ridiculous FATCA Act and the reporting requirements it imposes. Although I can't see why you would need one.