I love the Saarinen tables and chairs. You could go with the tulip chairs that were originally designed for the table. Design Within Reach and Knoll have them. If you can't spend that much, I am adding a link to some on eBay which are in that style which would work, too, I think. You could also go with Bertoia chairs which would blend nicely.

As far as the rug, it depends on what else is in the area of the table. I would go solid or maybe color block in a color you like or one that goes with other things you have. I would avoid beige or neutral since you want to break up the wood and a bright spot of color might be fun and work with the modern aesthetic you have.

Sorry.

Here is a link that might be useful: saarinen style chairs

I do like them, but I hadn't really considered the matching chairs because I thought it might be too much. I did find this picture where they look pretty great.
I want them to be very comfortable and was looking at some leather options as well. I am not sure how that would look though.

Our house is almost entirely neutral, so i would probably avoid a color for the rug. I agree it should be solid-I want the focus to be on the table.

Here is a link that might be useful:

The rug should probably be round and about 6' larger in diameter than the table so the chairs sit on it comfortably, so if you want plain, I would probably find broadloom carpet that you like and have it cut and bound.

Gorgeous table and floors! For modern, I would look at some of these stores.

Design Within Reach
Crate and Barrel
Room & Board
CB2
West Elm

I have thought about a large solid cowhide rug. I think the shape would work with the round table. I have one in another part of the house and it is the most durable rug I've ever owned-and I have two large dogs who play and sleep on it often.

I have a few chairs that I am considering. I would love some feedback.

Antica chair

Allegro Chair

Room and Board CHair

Here is a link that might be useful: Tulip Chair

Sorry, Here is the last one again.

Here is a link that might be useful: Room and Board chair

While I really love the red one you like, I think it would be hard to move those legs on a rug soooo I found this stylish chair that swivels. I think the style would go well and who doesn't love a swivel chair. Love the table.

Here is a link that might be useful: swivel chair

I think the tulip with the swivel would be perfect.

If your budget allows the Tulip, I would consider the Doyl at DWR: it's a bit more refined than the Antica or Allegro, imo. The Antica legs seem kinda spindly to me (?)

The R&B Lange is a decent knock-off of the Breuer Spoleto chair by Knoll at about 1/4 the price, but if you wanted to do anything different with the leather or use hide you could check out the real thing as well.

Here is a link that might be useful: DWR Doyl

I like them all, but my favorites are the tulip chairs or the Breuer chair at R&B.

I loved the Doyl in the picture, but tried it out at DWR and found it very uncomfortable. The Bottega was very comfortable and is another option. I've heard that DWR will let one take things on approval so when I narrow my choices, that would be helpful.
The R & B chair was more an example of the chrome/leather style. I've looked at the Spoleto, and the Mies van der Rohe MR and BRNO chairs. I would probably look for a vintage version of those. I'm not sure if that look is too much. I live in a 1900 home with a mix of modern furniture and some antiques.

Here is a link that might be useful: Bottega

If you want a vintage chair, you could consider Thonet chairs in black (1859)

Here is a link that might be useful: Thonet chair

I love Thonet chairs, but unfortunately my husband does not.

I am hoping to lighten up the area a bit. I think a rug will help. Which chairs would be best for that?

Sorry but maybe I stumbled upon the wrong thread but I have been looking for an affordable set of contemporary white dining chairs and came across this post.
I looked through many of the responses and to be honest, I like the chairs that you already have better than any of them. I think they would look great if you painted the chairs an off-white and sprayed the legs in a stainless steel or chrome finish. It would really brighten up the space and adding the cowhide rug that you mentioned would look even better. The white would look awesome with that.
I also think that some punches of lime green would look great in your space too...(Could just be that I am obsessed with that color right now but I find that it looks great with almost everything).
In case your interested in some that I have been considering, I've provided a link to my Ideabook where I have been saving a few...Prices and info included on where you can get them and the prices are much better than what I have seen here. (I personally don't like the idea of the tulip chairs...it's too much).
Here is one of the chairs that I have saved but pardon the bad image...having problems with my photo editor.
Good Luck!

Here is a link that might be useful: Contemporary white dining chairs