Without a doubt, check out
www.alojamientos-sevilla.com for apartments. Last year we rented one of them for 4 and a half months, but they are usually rented out for much shorter periods--even for a weekend. Antonio Bomba is the proprietor and he is such a sweetheart. There are a good selection of apartments in the Barrio Santa Cruz, but I don't see it on the website now. There was some water damage from a broken pipe in the apartment above, and they may still be doing repairs. Too bad. It was beautiful. These are not budget accomodations. However, depending upon the length of your stay, some of them could compete quite well with area hotels. Good luck!