I just got a call from 801-945-5000. I did not answer . They left message saying American Express and to call them back at 866-379-3443.
I then got an email right after stating that my card had been approved . I had applied a couple days before. The email was genuine from AE .
So this number is indeed from american Express .

I received a call from this number and they left a message.  It would be a lot less of a mystery if they would reference why they are calling on the message they leave.  Until they do, they aren't getting a call back from me, even if it is amex.

I have an online savings account, but no cc with amex.

The 801-945-5000 is a real american express (amex) phone number.  The thing is, with the new caller id fakers, people can call you from any number and choose the number they would like to show in the caller id.  When the people called they left another number and a reference number to call on my voicemail.  They probably use the real amex phone number in the caller id to trick you into answering the phone or they leave another number to call on the voicemail message. The reason why I found this suspicious is because they also called my grandmother's home asking for me and I've never given amex my grandmother's number. I called american express and they confirmed the 801-945-5000 as being an office that handled their gift checks.   I also called the 866-379-3643 number that they left on my voice mail.  I gave them a fake name and year of birth and they said the found me in the system.  Then they asked for my full social security number (SSN) because they said they could not see any requests made for me and needed to check with my SSN.  I'm almost 100% sure this is a well executed scam to steal your identity.

I haven't applied for a new line and still received the call. I told them I have been a customer for over a decade and all of my info should be on file.

Moreover, I have been getting allot of spam calls and even text messages. This is why i was so suspicious.

Got a call from this same 801 number, actually 2 in 20 minutes. Caller ID listed number and "Private Name". Caller left msg 2nd time and said that she was with AmEx security and wanted to verify my 4K $ purchase of gift cards. I called # on Amex card and they sent me to security. Caller (and number) was legit. They cancelled the fraudulent charge, invalidated my card, then confirmed that a new card would be delivered tomorrow.

Bottom line, though someone could fake the number on caller ID, in my case it was valid. If you receive a call from this number, and have an American Express card, call the number on the back of the card, speak to a representative, and ask them if they are looking for you about an issue with your card. If you dont already have a card, or have applied, Google AmEx's customer service number, and give them a call.