It might be possible to move them, but you need to dig up as much of the roots as possible. I have successfully moved a few, but expect transplant shock and almost sudden death, unless the area they will be in has some really good soil. b sure to water very well every day for about a week, so they can perk back up again.
Good luck

Better yet.... can you tie-up/or prune the tomatoes away from the basil? I have a zucchini plant that is trying to over-take a jalapino pepper, so I just keep cutting off the leaves that are near the pepper.

Transplanting basil usually goes pretty well but you might just consider reseeding if the plants have beciome leggy from being overshaded by tomato plants. I plant the two beside eachother but always locate the basil on the sunnier side. I pulled radishes and reseeded and the new seedlings will soon cath up with the older planting.. if I can thin them a little.

You might have better luck if you chop the basils right back, and use them as cuttings to make more babies to put elsewhere. Basil grows very easily from cuttings - I prefer to use the somewhat older bits, instead of the tips, which are very tender. If you can get a heel cutting from a woody part of the plant, that will be the best way of doing it.

Tip: when taking cuttings, make sure to cut the stem on an angle, not straight across. That gives more area for the roots to form. That's why I prefer heel cuttings whenever possible, too. They strike better.

The problem is that I have such limited space that I think I might have to put the existing plants in pots and obviously, potting soil. Would that be a problem?

I have done both to basil. Moved a whole plant about a foot, and propagated via cuttings twice. Of all the plants I have, in this my first season, Basil is the most vigorous. My basil was towering over 3 feet until I harvested a bunch for pesto.

I think with a little care (as mentioned by others) you will not have any trouble whether you transplant or just propagate cuttings.

Good luck.

Cheers!

bmoser that is one incredible row of tomatoes and basil!!

I'm assuming you run a commercial garden/greenhouse?

It really looks awesome.