@ the OP: one thing is, what's offshore to you i.e. what country are you in?
In my experience (UK) I have yet to see an organisation who was satisfied with work outsourced to developing countries (pun not intended). Most ended up abandoning their stalled projects part way through, losing money.
There will always be someone cheaper that you. The thing is to decide your approach, your model, and what strengths you will play to. Reasonable price is one factor, but only one of several - e.g. quality, reliability, etc.
Some clients do look only at price, mainly for small projects, but I always left them to it. Not worth chasing.
In 8 yrs I've never had to ask for a deposit, and have never been done over, except for a tiny amount when one place went bust. I think that comes from not chasing anything and everything, the result being decent projects and decent folks to work for. Or it could be blind luck 😉