You wont know till you try, but since its in the EU you should be able to travel with a photo driving licence. I read about some bird before Christmas - had lost her passport but had a pic of the business page on her iPhone. They let her travel. Good luck.
PS: I think it is the airlines call since they have to pay to ship you back if you are denied entry at the other end. British Airways might be your best bet?
Your passport must be valid for the proposed duration of your stay. No additional period of validity beyond this is required. However, it is always sensible to have a short period of extra validity on your passport in case of any unforeseen delays to your departure. You do not have to wait until your old passport expires to apply to renew it. Any time left on your old passport when you apply will be added to your new passport, up to a maximum of nine months.
Citizens of the European Union (EU) member states and Switzerland can travel to the Canary Islands with just their national identity card. Nationals of the UK have to carry a full passport (UK visitor passports are not acceptable), and all other nationalities must have a full valid passport.
You also need to provide Advanced Passenger Information (API), you will need to provide to your airline, either at check-in or more probably when you made your flight booking, the details contained on your passport photograph page such as your full name, date and place of birth and passport number, or was that when you noticed it had expired!
From June 1st, 2008, API, advance passenger information, will no longer be accepted at UK departure airports. This means you must either supply the information when you book or use your airline's website to do so at least 4 hours in advance. The Spanish government has required API for all air travellers to Spain since June 1st, 2007. The advance passenger information requirement applies to the whole of Spain, including the Canary Islands.