Based on our conversation with the IRS (through their business helpline, 800-829-4933), we do not believe that the IRS rules would prohibit a bank from linking HSAs to savings accounts for purposes of overdraft protection. Unlike an overdraft transaction that is covered by the bank, such transfers would not be considered loans to the HSA account (which are prohibited transactions, see Question 35, Internal Revenue Bulletin, Notice 2008-59 (July 21, 2008).

However, any transfers from the savings account to the HSA would be considered contributions to the HSA account, and they would count towards the account’s maximum annual contribution limit. And, as you pointed out, each account could have a different limit for annual contributions. The bank would have to monitor accounts with linked savings accounts for excess contributions, in addition to monitoring for prohibited transactions (see Question 68, Internal Revenue Bulletin, Notice 2004-50 (August 16, 2004)). Also, your account agreements may impose additional monitoring and notification responsibilities. 

The IRS representative noted that beneficiaries might be able to avoid tax penalties for excess contributions by lowering their contribution levels to leave a cushion for overdrafts or by returning any excess contributions before the due date for their annual tax returns. See IRS Publication 969.