How to ask for an earlier start date

Last Saturday I had an an interview with a company at a conference and was offered a job the same day. During the interview I remember explaining my current situation and that I could start as soon as possible. The phone call from the recruiter congratulated me and I was offered the position while she stated both the salary and relocation bonus pay. I did not hear anything about a start date. I was required to fill out an employment history application first before they send me an offer letter. I should be receiving the offer letter this coming Monday. My question is should I ask about it now before I sign an offer letter and is it negotiable to ask for an earlier start date if it's not what i'm looking for?

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