As a lender, I get asked this question all the time. Also, there are a number of polls out there saying that 80%-plus of millenials think you need a large down payment to buy. This is not true. You need something but not much. FHA has the 3.5% down program with a pretty major Mortgage Insurance tacked on each month. But, as another respondent stated, banks (and more specifically wholesale lenders available only through mortgage brokers) have begun to get more creative with their programs. For instance, I have a wholesale lender that allows a 1% down purchase to which they will GIVE the borrower an additional 2% themselves to make a 3% down payment total. THis program is available either with Mortgage Insurance (cheaper than FHA's, by the way) as well as with Lender Paid Mortgage Insurance which is reflected in a slightly higher interest rate than what you might have qualified for otherwise.
Above all, do NOT limit yourself to a bank for your mortgage needs. Banks only offer their own programs and will never tell you about another lender's (possibly superior) product. If they can't fit you into their own box, they will just turn you down, oftentimes leaving you with the loss of an application fee, a credit pull and,many times, an appriasal fee. Check with a reputable mortgage broker in your area. You'd be surprised at how many more flexible products are available through that channel.