Great responses to your question. I would just like to clarify a couple of things: Whether with or without a formal contract, whether W2 or 1099/self employed, regardless of how long or short you've been with your current employer, you always need to show a 2 work history via W2s and/or Federally tax returns. Only exception is if your fresh out of college and been employed for less than 2 years as a result. In that case, however, so long as you're employed in the line of work you went to school for, your time in school counts towards your work history.
Your situation may have other variables not stated so talk with your Loan Officer about the specifics.
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However . . .
I don't see why not. After all, people who are "employed" without a contract are what's called "at will" employees. That means that the employer (or employee) can end the employment arrangement at any time. There's no protection at all. (Sure, there are EEO and other protections, but there's nothing saying that if you work at the local flower shop or at a gas station or as a secretary/administrative assistant in a big company, that you can't be terminated tomorrow. And usually without cause: You can just be fired.
A contract at least is clear about the length of employment and the terms under which it may be ended.
So, see a lender to find out about financing in your own situation.
Hope that helps.
Best of luck,
Keller Williams Realty