I think some notion of make, model, year, and a description of the
attachment points and what's wrong with them would be good here (dare
we hope for some digital-camera snaps?)  Bumpers have been made of a
lot of things and attached in a lot of ways over the years.

Let's start with what's there now:  Two big metal prongs with bolt
holes in them?  Things that look like horizontal shock absorbers?
The rusted remnants of such things?  A shaped mass of high-density foam
whose hard plastic exterior has popped off like shelling an M&M?   What
kind of shape are they all in?  You get the idea...

Cheers
--Joe

No, it is not something you can do. You are looking at the bumper and
asking strangers whether you can handle it. This always means no.

Either the bumper or its mounting was badly rusted out or there was
more damage than you observed. In either case, it is not simply a job
of reattaching the bumper.
If you can find a place for some bolts there, why not go down to the
lumber yard and purchase a 2"X12"  of sufficient length for the
project. Bolt the new bumper on with the studs facing forward. That is
very important. You want the next person that hits you to suffer the
maximum pain. Paint the bumper silver or to match the color of your
car. Even though your car is somewhat older, there is no need to look
tacky.