Q: What banks offer non-conforming (portfolio) mortgages?
A: Most banks have some kinds of portfolio lending that they will do.
For mortgages, these largely take the form of short-term loans, some adjustable-rate mortgages and jumbo mortgages. That said, it should be noted that "non-conforming" covers a lot of ground, since it specifically means "cannot be sold to Fannie Mae or Freddie Mac," and isn't necessarily the same as "portfolio."
"Conforming" is a definition, and simply covers a set of standards or requirements: Credit score, down payment, documentation, asset strength and collateral. Anything which does not meet any or all these Fannie, Freddie requirements would generally be non-conforming.
In the market today, a portfolio loan--one the mortgage lender writes and accepts for its own investment book (portfolio)--may have standards as rigid or even more rigid than a conforming loan.
However, when writing for their own books, a lender might consider offsetting a weak standard with an ultra-strong other one. For example, a weak credit score might be offset by a much larger down payment. In such a way, they can balance the risks of making a loan for their books. In such a case, the loan would be a non-conforming portfolio loan, since it no longer meets Fannie or Freddie standards and is held by the lender.
A lender could also keep conforming mortgages is his portfolio, too.
Have you considered a mortgage broker?
What a lender will accept for its portfolio is highly individual to the institution. If you are looking for something very specific--like an interest-only loan, or one which allows very high debt loads relative to income, etc.--you'll need to contact individual institutions directly to see if they will or will not be able to meet your need (or if so, under what set of terms and conditions). You might also consider engaging a mortgage broker, who may have ready access to lenders willing to meet more unusual needs.