I was terminated from a job about two years ago. I am currently
pursuing another career and the company is doing a background check on
me. Are their any circumstances where my old job can notify a
potential employer that I was terminated.

Most companies are unwilling to disclose anything more than dates of
employment and possibly confirmation of a salary range or eligibility
for re-hire. I don't think the re-hire part is common and some
companies won't rehire anyone that quit so it doesn't mean you were
fired. Typically any company today is at fear of risk of lawsuits over
negative info or even positive info if someone is recommended and
turns out to be a bad hire. So usually they'll disclose as little as
they need to.

In my experience you can't be sure if old employers or references
would even be called. In many cases personally I've found they haven't
called. In one case I had a new manager tell me that they got some
questionable feedback from a previous manager but I was hired anyway.

I suspect it varies a lot depending on the size of the company. If
there is a large HR department expect the minimum. If it's a smaller
employer there's more chance they may be less discreet.

I'd suggest having a friend at some other company call for a reference
or background check, using a real name and company phone so that they
can leave a message for a callback if needed. See what they get told.
If you find negative info is being given then I'd contact the former
employer and politely express concern and ponder out loud the
legality. Don't threaten but let them know you're aware of the
negative info and are concerned, maybe saying "I didn't think that was
legal to do that". Advise them you are looking for work and are
concerned about future calls. This may cause a more guarded reply on
the next call.

I'd avoid lying on applications. If it gets to the point you have an
interview and you know references will be called it may be worth
describing the circumstances of termination in the best way you can
(mutual agreement it was time to move on). Avoid overly negative
bad-mouthing of the previous employer, but it's ok to admit you had
differences of opinion or found that it wasn't the right job or place
for you to be working at.