TALLINN, Estonia—An Estonian court is opening a hearing next week that is attempting to challenge the legality of the euro zone's permanent rescue fund, just months before the crisis-fighting mechanism is expected to come into force.

The Estonian Supreme court could take up to four months to issue a ruling on whether the treaty establishing the European Stability Mechanism violates the country's constitution. But the verdict could set a precedent for other groups challenging the legality of the fund, which is meant to shore...