Italian prosecutors and the Carabinieri, the national military police force, which has a division that investigates art theft, have opened investigations into the painting’s movements since it was taken from Italy. They have asked German judicial authorities to cooperate.

They are also evaluating whether the family’s request for money could be construed as extortion.

“What belongs to the Italian state has to be returned to the Italian state,” said General Fabrizio Parrulli, the commander of the Carabinieri’s art theft unit. He said his men were working with Florentine prosecutors on the case, but declined to give details, “because the investigation is ongoing.”

In such cases, German officials have said the 30-year statute of limitations means that when property is in private hands, there is simply no legal way to force its return, and no basis for government intervention.

For many years, Israel and Jewish groups have lobbied Germany to carve out an exception for items looted in the Nazi era. After the discovery in 2012 of more than 1,000 artworks in a Munich apartment, including some that were taken by the Nazis, the government considered such a change, but it never became law.

Mr. Schmidt said that generating publicity about “Vase of Flowers” should make it harder for the Germans who hold the painting to try to sell it.

“Thanks to the photo in the Room of the Putti, people will never forget that this work was stolen,” Mr. Schmidt said in a telephone interview. And thanks to the accompanying resonance on social media, “no one would ever be able to say that ‘I purchased this work in good faith,’ ” he said.

But some experts say that Italy should examine its own record on stolen art.

Italy was recently called out at an international conference on the restitution of art looted during World War II as one of five countries that have been slow to address the issue.