Unfortunately, it is pretty hard to have a contract for Fiber Internet connetion with most of the providers without a NIE card. I suggest you to choose Movistar Fiber Internet without contract package.
If you need a extensive mobile data plan, I suggest you going with Simyo which uses Orange as backend and provides 25GB 4G Internet connection for €28. In Movistar mobile, you can get only 2.8 GB with this amount of money. You can visit a tienda (shop) and get the SIM card for €10. Then you can load Internet and call packs on the control panel. If you don't have an existing Internet connection, you may ask for the representative.
We came to Madrid, Spain to work for Amazon and applied for our NIE card on 25.07.2017. Our appointment had been arranged for us before our arrival. We were told it may take 1 - 1.5 month for the card to be ready and given a stamped document indicating we've applied for the card and currently on hold. This document was supposed to be a temporary NIE card. Because it contained the essential information like expiry date and NIE number. We also had our NIE numbers on our visa (passport page).
Applying for Internet Connection
After finding a flat in Madrid, we applied for the Fiber connection and 2 mobile lines (to be moved from Movistar) on Vodafone's web site on 06.08.2017. They called us back the following day (07.08.2017) and asked the same information again on the phone (the call was in English). Then I received an email indicating another application was registered than the one I applied the previous day. On 08.08.2017, someone else called me and told me that Vodafone asks for €500 as assistance fee and it would be better for me to go to a tienda (physical Vodafone store). I went to a tienda in Delicias and they told me to wait the men who will come for the installation and they will bring the contract. But I didn't trust his words. The next day (09.08.2017), I went to the tienda in El Corte Ingles, Paseo de la Castellana. I provided my temporary NIE document, my passport / visa copy, my work permit. The saleswoman asked her boss and the next day (10.08.2017) told me they can not accept the application without physical NIE card. So I went to Orange store near Puerta del Sol. They told me the temporary document is invalid and they can not make a contract! So after recommendation of my friend, I went to Vodafone in Puerta del Sol and they accept the documents I provided and take notes on it and sent me because they were closing.
The other day (11.08.2017) Puerta del Sol tienda of Vodafone called me and asked me my IBAN number (for automatic payment). I provided the number and could not get a response until 16.08.2017, when I received a call in Spanish, saying I have to go to the tienda to sign the contract. So I went to the tienda in Puerta del Sol. After waiting for 40 minutes, I was told my contract was cancelled by Vodafone due to the need of €500 deposit. They cancelled it without giving me a chance to pay the deposit (which is kind of insult for me). Also there were 4 applications on behalf of me and all of them were cancelled. I lost many hours dealing with these problems.
It's so strange that the modem they'll be giving doesn't worth €500 and I'll be paying the bills every month and I work for Amazon with indefinite contract.
On 20.08.2017, I applied for Movistar online, they asked for NIE and IBAN. The next day, they called me and told me I had to reside in Movistar for 45 days to apply for a contract and they could offer me Fiber-only package without contract. I accepted. On 23.07.2017, the technician came and installed the fiber cable and the modem. They even didn't ask for my passport or NIE card! It's now working properly but asymmetrical (50/5 Mbps) for now.