Delta and United both have low fares in the mid-to-upper $500s for Phoenix to Barcelona and Madrid. Delta has many other cities around Spain on sale too. Lowest fares are travel departures January through March or early April with fares rising to $600s in April and May.
United/Star Alliance $548 – $574 January 8 – April 5
Delta/SkyTeam $552 – $584 Jan 9 – April 4
United/Star Alliance $549 – $572 Jan 9 – April 5
Delta/SkyTeam $556 – $585 Jan 10 – March 21
Delta Airlines Phoenix PHX – New York JFK – Barcelona, Spain BCN
$551.26 round trip Wed March 1 – Fri March 10
Google Flights fare map for Phoenix – Spain
SkyTeam Alliance Flights Sat Feb 4 – Sun Feb 12
Fares I checked on Google Flights show low fares, however, they do not include a deep link directly to booking site for tickets outside Barcelona and Madrid. My searches on KLM showed fares in upper $500s to these locations.
Skyscanner.com and Momondo.com are useful for pulling up online travel agency low fares to these destinations in Spain without direct booking links through Google Flights.