Is it possible to attach an HSA to a current health insurance policy or do you have to buy a particular plan to get these benefits?

This is a good question. HSA, health savings accounts are considered qualified plans. The way these must be done is you must have a HSA qualified high deductible health insurance plan. This could be an individual HSA health insurance or group health insurance. Either way it has to be a qualified plan. So, the short answer is No you cannot just add a health saving account to any health insurance plan you might have.

if it fulfills the conditions HSA, you need a particular plan for these benefit

The health plan has to meet the requirements for an HSA - primarily to be be a high deductible plan.

If the plan you have is not HSA compatible, you may be able to change your plan with no additional medical requirements to one that is. That is what we did. For us, the deductible stayed the same ($3500) but the premium is less.