ENBD, RAKBANK, Dubai Islamic, Mashreq.
If you don't have a EID the above only open for private bank relationship.
Citi, HSBC, Standard
It you already have some link to them outside the UAE.
In general if somebody gets his EID through company sponsorship we open always try business account first and with the same bank meeting making the application for the personal account so that it makes sense for the bank to open the personal account