Im afraid it depends on the frame. I have a bike that takes 26 x 1 & 1/4" (597mm diameter) rims that will just about fit a 27" wheel, if you deflate the rear tyre before putting it in and then mudguards will not fit with a 1 & 1/8" wide 27" tyre.
The front fits fine and still has a decent amount of clearance. 700c would be slightly better, having 8mm less diameter.
Another option is to fit 26 x 1 & 3/8" (650A - 590mm rim) rims and tyres, but again you have to watch the clearances, especially with the fatter tyre. On the above mentioned bike, the clearance is a little tight and, when climbing out of the saddle, the tyre will rub on the chainstay. Tyres and rims are still readily available in this size. Michelin World Tours are the thinnest tyre I've found in this size.