You will finad this info vary rarely CHEAP shopping and souvenirs in dubai. Yes it is possible. there is a place called Al karama market which is near the inner city deira area. you would have to ask the cab driver to take you to the Al Karama Market.
There is the Al karama centre as well but always go to the market first. Karama is a south indian quater of the city where all the south indians live and the markets are full of souvenirs and little things you can buy to take home. i bought a crystal Burj al arab for 10dhs - in the malls and tourist area at least 100 dhs
remember to haggle and bargain these are indians remember!