When you say 'previous bank statements', is it existing mortgage statements for a refinance, or is it savings statements to demonstrate genuine savings for a purchase?
In both cases there are lenders who don't need to see statements in certain circumstances, but not all.
If you're worried about genuine savings, there are a few lenders who will lend up to 90% without seeing your savings history. They will need to know the funds are available though (an account summary will show this).
There's also a couple of lenders who don't need loan account history to refinance, but they do tend to be a little more expensive than other lenders. Also be aware if you're worried about past account conduct, your credit report may also reflect some things.
The best solution is to speak with a broker about your specific circumstances as general advice probably won't help you.