Sure. The owner is the owner until the title is transferred.
However, there are a number of problems with that. First, the rental income is income . . . and that should be reflected on the financial documents the seller is providing to the lender to demonstrate hardship. If it's not included, then the seller may be committing fraud.
Second, it's highly unfair to unsuspecting renters. It's one thing if the owner tells the renter: "I'm in financial distress. I'm trying to sell this home as a short sale. If I'm not able to, the lender might have to foreclose and you'll have to find another place to live." But if the owner doesn't reveal that information, that's really bad behavior.
Third, it'll make the short sale property more difficult to sell. Generally, when a property is sold the lease continues in force, applicable to both the tenant and the new owner. (The lease can be written so that it's terminated upon sale of the property, but most aren't.) Suppose a tenant is 4 months into a 1-year lease when the property is put on the market. A potential buyer would be faced with the prospect of buying and then having a tenant in place for half a year or so. That's enough to discourage many buyers.
So, the answer to your question is "yes." However, it may not be the best decision for the seller.
Hope that helps.