The Massachusetts millionaire’s tax is back on the table—tweaked to avoid violating the state’s constitution.

Two bills filed Jan. 18 would amend the state constitution to place a 4 percent surtax on income over $1 million. The proposal could generate $2 billion a year in new revenue to be used to fund education and transportation initiatives.

But the tax faces a long road.

The earliest the measure could appear as a ballot question would be 2022, because it needs approval...