With a bob and a weave and a joke — and a lot of winking — former Chicago Public Schools chief Paul Vallas on Monday made it clear that he is interested in running again for governor of Illinois.

In remarks before and after a luncheon of the City Club, Mr. Vallas, repeatedly insisted that, while he would never say never to a race here, his hands are full with his current job as superintendent of schools in New Orleans.

But such insistence quickly crumbled under persistent questions from reporters and from Mr. Vallas’ own hints that he really wanted to say more.

“Would I be open to running again? Yes,” he ultimately concluded. “Am I planning that? No.