I have copied the following 2 points from a mail list for Arabic teachers. (I hope they don't mind). Perhaps there are some useful points. Good luck!
1.
Many foreign students residing in Damascus stay in the Bab Touma area of the old city, where there are quite a few families willing to host students. Most people I know who've done this have gotten a lot of practice with their colloquial Arabic. Your program may be willing to set you up with a specific homestay, but again I don't think this is entirely necessary.
Price in the area vary from about 7-12 thousand Syrian pounds($US140-240)/mo, though one should always bargain. This can as much as halve the prices on some rooms.
In order to look for rooms, you can simply ask shopkeepers, etc. Internet cafes can be quite helpful as well, particularly since the staff often speak excellent English if that is an issue. It is quite rare for someone to look for a room more than a week or so in advance, and if you do you may be obligated to put down a deposit or begin paying rent early.
Be sure to inspect the kitchen and bathroom facilities for renters, which are often separate from those of the family, and make sure that there is a "khazaana", a storage device for water, as it generally cuts off in the afternoons.
2.
The two most popular destinations are:
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University of Damascus
Arabic Language Centre
Language Institute
Damascus
Tel : +963 (0)11 2129494
Fax : +963 (0)11 2120464
e-mail : inquiries@arabicindamascus.edu.sy
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Institut Français d'Etudes Arabes de Damas (IFEAD): http://wwww.ifporient.org
You may also wish to consult the Syrian Studies Association site which provides detailed information and reviews of the various places in Syria that offer Arabic classes: http://www.ou.edu/ssa/learn.htm <http://www.ou.edu/ssa/learn.htm>