It doesn't look like those are original..wonder if they are hiding something.
I would get a putty knife, see if you can slip it under and then tap lightly with a hammer to see if you can get them loose. you might be able to chip them off. Start at the bottom so you don't ruin the curly cues.

I'd leave it alone. Unless you find fastners on the back side of the door, these circular pieces are likely glued on. If so, prying these off is likely to pull splinters off the base wood making a mess of the doors.

Personally, I think the round pieces are ok. The curlycues are sized to rest on the top and shoulders of the round pieces and these seem to go together. Removing the round pieces would leave the curlycues dangling in space with a sort or slapped-on look.

If you or a friend is handy with "Photo Shop" (or similar), try this: Take a digital photo slightly off axis so that the edge of the round piece shows, and then process a copy of the photo removing the round piece. This will give you a preview of how the peice may appear after removal of the round pieces. Compare to judge if removal improves appearance.

How about adding to the "snowman" circle instead of possible damaging the piece by taking it off?

Monogramming comes to mind. You could use a single, or multiple, depending on the initial that your last name starts with.

http://mintjulepmonograms.com/wooden-monogram.html

Another idea would be to add more "curlycues" or scrollwork. See link below.

Or, you could simply have something painted on the circles.

Here is a link that might be useful: Appliques and Overlays

If you are planning to paint the piece anyway, I would just sant the edges of the circles down first to make them disappear. Easy peasy.

How old is the piece? If it is antique, you will ruin the value by painting it and removing original parts.

Christopher

Good point. Best to get a value on the piece, first. TA good dealer would be able to tell you if the circles are original. If the piece isn't worth much, have at it! If it is valuable, you might consider selling.

The wood - the part I can see - looks too good to paint. Is it walnut? I would buy a carved wood applique and place it on the "snowman" to make it's outline less obvious.

Question. I notice you say it will fit into your bathroom perfectly. Am I jumping to conclusions that you are going to use it for a sink base? I see a lot of people doing this to antique furniture. If that is your intention, worrying about preserving its value is sort of moot. If you are going to use it for something like linen storage get familiar with ways to protect it from water or moisture damage in that environment. I also agree it may splinter the wood beneath if you try to remove it. You can buy wooden appliques and glue them on to remove the snowball appearance, but again that would make worrying about value a moot point too. Consider just rejuvenating the finish a bit, since it appears to be in good shape and not painting. I find it really looks richer than painting. People paint antiques when they think they're not worth restoring.............this piece doesn't need hidden. It's pretty.