I would go with a neutral tone on tone sculptured rug, almost off white in color. The chairs are beautiful but they are reading pretty close in color to the hardwoods. I think breaking them up with the same color as on the dining chairs would really set them off. I think the navy pillows would be excellent and tie the art in the dining room into the living room. Maybe add a small table in between the chairs and use blue accessories as well. I would also add a floral as a centerpiece in the dining room that is the color of the chairs. It will just help to tie the combined rooms together.

Not sure if you like a modern look, but this rug from Crate & Barrel picks up on your wall and chair colors, and navy pillows would certainly work nicely with this color combo. I also like how the pattern of the rug is similar to the one on your chairs, only on a slightly larger scale.

Planning on putting this table between chairs. Thank you so much for the help!

love that table. I really like the rug that karemore suggested but I am just afraid that there would be a LOT of pattern going on. Might be too much for the eye to take in.

You need a bold color and design to set off the two chairs. There are a lot of contemporary rugs on overstock.com, rugstudio.com, rugsusa.com, and wayfair.com These are just some examples

Or here is a more traditional rug from Pottery Barn. Also picks up on swirly pattern of your chairs. Navy pillows would not look right though. You would have to pull a different color from the rug for your pillow color.

How about this one, also from Crate & Barrel? Still too much pattern? With this rug, you could go back to your idea of navy accent pillows.

I really like that one, but I am just scared about mixing such strong patterns... Look for a tone on tone one

If you're thinking about navy pillows... a rug with navy in it?

I think your room could handle the pattern of the striped rug with the chairs. There isn't a lot of other pattern going on in your space, so I think it would be a clean look. I feel it's important to bring in some more red or the chairs will look a little out of place. Is it color or pattern you're worried about?

We just started working on this room after having the hardwood flooring installed. What you see is basically the whole room; it is a living/dining room. Thank you so much for the comments so far. They are really helping.

You chairs need a rug that mimics the two shades. This one will do it. Navy is great for an accent.

I love carolins suggestions either would look great depending on whether or not you want something traditional or more modern.