I need to open a US bank account for my company (it's a swedish
company), but I can't find any US banks that accepts non-US companies
as customers. I should be able to transfer money from PayPal to my
bank account and I should be able to send wire transfers to non-US
bank accounts (to accounts outside US). It's also a plus if they offer
Internet banking so that I can check the balance and transfer money.
If they have a minimum deposit it should be less than $1000.

Are the U.S. banks telling you that you need to
have a U.S. presence before they can do business
with you?  Kind of makes sense.  Also, check out
the internet banks, not sure of there names.
What reasons are the banks telling you?
Ask them next time you get rejected.

Ideally, you need to be introduced.

Many years ago, as a UK national domiciled in the UK, I opened a bank
account in RI ... largely because I knew one of the VPs.

Also, you may be able to form a US Corporation but, again, an
introduction could help. (The company that forms the corporation may
provide this service)

Hi there,

I need something similar for myself, so I spent quite some time
searching. The only site I found that appears to be legit and
reasonable is Valis International. For a personal account they meet
your criteria, but a business account requires too much in up front
fees.

Personal
http://www2.valisinternational.com/?pg=personal

Business
http://www2.valisinternational.com/?pg=business

robertskelton-ga

Hi Bylla,

Gee, I was going to tell you about Valis International,
but robertskelton-ga already has.

He's right. I have clients that use their services.
And they are very happy with them. 

Best wishes,

Your TaxMama-ga