Gov. Mike Dunleavy told legislators in a pair of closed-door meetings last week that he will introduce a state sales tax as a component of a budget-balancing, long-term fiscal plan.

But with just three weeks left in the legislative session, with no details about the governor’s tax bill as of Monday, and with strong opposition from lawmakers who represent communities with a local sales tax, the odds of passage this year are extremely low.

If the governor goes ahead with a sales tax bill, it would join more than a dozen proposals offered by House and Senate members this year to address Alaska’s...