Does bank interest paid offset interest received?

No - interest expense is reported separately form interest income. One reason is that interest expense is an itemized deduction, so you could have interest income that is taxable, but not deduct any interest expense if you don't itemize because the standard deduction is more than your total itemized deductions.

Also note some interest expenses such as interest paid on a credit card or line of credit (other then possibly a HELOC) are not deductible anyways. So your interest expense may not even be deductible.