It's been like 4 years since I had to do this, but this was how it played out for me:
- Bank of America closed my account which left my biggest Paypal at the time with no bank account to withdraw to
- It was a Paypal I bought from here which I never bothered requesting the original sim card or phone for, so I was in the same boat as you (no access to phone)
- I got the quick security check when trying to add a new bank account to withdraw to
- Couldn't get around it because I didn't have the phone they wanted to call or text
- I went into my account settings and added a Google Voice # that forwards to my personal cell phone
- Tried to add bank again, got QSC, no luck, new number wasn't showing as an option
- For the next 4 or 5 months (not sure, but I know it was a few months) I used all the money that came into that Paypal to pay my supplier. Didn't withdraw any money because the only bank account attached to my Paypal was a closed bank account.
- Not expecting success, and mostly for sh!ts and giggles, I tried adding the new bank account again several months later (again, not sure exactly how long, but a good while), got the same QSC, but to my surprise, the Google Voice number I'd added to the account now showed as one of the options to receive a verification call or text.
- Told Paypal to send the verification text to my GV #.
- Nothing showed in GV inbox
- Told Paypal to resend the verification text to my GV# because I wasn't seeing it
(note that this has always been a weird glitch I experience when trying to get verification texts through GV. Almost every time I have to request the code be resent, and then both codes show up)
- Both verification texts hit my GV inbox after I request the 2nd code
- I put the most recent verification code into Paypal
- I add the new bank account
Might work the same these days, but I unno.
Question about Paypal "Quick Security Checks"... 
