I'd be surprised if you could and be satisfied with the result.

What have you used to try to clean it ?

Does it feel rough to touch or is it just discolored?

If left soaking in liquid bleach, it is likely the flat metal around the drain will corrode but the drain likely can be changed. You might, however, consider soaking the stain in a paste of powdered cleaner containing bleach such as Comet or Ajax. You might also try circular motions with steel wool soap pad on a corner or edge where it would not be obvious to see if that would scratch the surface -- if it doesn't scratch it, it may well clean it..

Do you know what material the countertop is?

An afterthought : You might try protecting the metal around the drain with tape or wax or even nail polish and decrease the likelihood you'd need to replace the drain if the Comet or Ajax paste runs.

Another afterthought: Measure the top edges of the sink (at the countertop) so you know the size and google stainless steel bar sinks that size. You might not be able to put a "kitchen" sink within the opening after cutting out the existing sink but you might find a bar sink that will fit the space closely enough, perhaps even one with the drain in the center.

https://www.amazon.com/dp/B004038JE2/ref=sbl_dpx_B0163D3I7O_0

It looks like a solid surface countertop with an integrated sink. if it is, you can actually sand out imperfections/stains/scratches with a mouse sander and fine grit sandpaper!

edited to include a link with more info: https://blog.solidsurface.com/4-tips-to-sanding-corian-guidelines-for-a-perfect-finish/

Looks like hardwater stain before u sand. Try some CLR snd a non scratch pot sponge. My bathroom sinks use to get like this. I tried bathroom cleaner even Clorox but the CLR worked. I closed drain pored enough to cover brown and let set for about 30 min then scrub till clean.

If the counter and sink are Corian you can sand it out.

https://www.corian.com/IMG/pdf/k-26840-corian-solid-surface-care-and-maintenance.pdf

I use OxyClean on my solid surface sink. A hot water soak for an hour keeps it looking new. Maybe it could help…

I'd have that looking new in about 20 minutes.

Random orbital palm sander, 80, 120, 220, 400, red Scotch Brite protocol.

Try using "the Pink Stuff" I am just amazed by what it gets clean. I used it on my rusty 1920's bathroom sink (I know I need to replace) and it came so clean. It may need some elbow grease but, it really works.

If it is Corian I bet you could have it cut out and have an undermount sink installed, but it should also clean up too.

Try Bar Keeper's Friend--the gold can. I would also try Krud Kutter (the red and white bottle).

I would listen to Joseph Corlett, LLC, above!

I appreciate all the cleaning advice - but I would really love to replace it with a stainless steel sink, which is so much more practical. That is what I am trying to figure out.

Joseph Corlett can give you the low down on that, as well. There’s probably threads on it as well.

I'm betting that top is Wilsonart's discontinued Solid Surface Veneer. You can tell by the skinny line at the top of the bowl. If so, you'll have to cut that sink out and top mount another.