If you're talking about playing Renaissance lute, then an 11-string alto guitar might be what you want. Heikki Rousu makes some nice ones. Google his name. The 11-string alto guitar is tuned up to G and uses a regular set of strings for classical guitar plus the lower basses use a 6th string from a regular set. The shorter scale allows for tuning it up to G.
If you're thinking of playing Baroque lute then you wouldn't want a high A for the 1st string, you'd want 1=F, a half step higher than the usual E. You'd want someone to build you a guitar to accommodate the D-minor tuning.
By the way, I have experimented using a high A on my 650mm 10-string. In order to do this you need a really thin (small diameter) string and it sounds crappy. Even G sounds marginal.