In a TV interview, Mr Putin said President Alexander Lukashenko may have made the threat in a fit of temper.
Mr Lukashenko is facing new sanctions over a growing migrant crisis at the country’s western border with Poland.
EU officials have accused Belarus of provoking the crisis to undermine the bloc’ security, which it denies.
Thousands of people, mostly from Iraq, Syria and Yemen, are at the border with Poland, enduring freezing conditions in the hope of crossing into the EU.
On Saturday, Poland accused Belarusian soldiers of trying to cut through the razor-wire border fence so migrants could pass through.
Earlier, Polish police also said the body of a young Syrian man had been found in the forest not far from the border in the same area. At least eight migrants have been found dead on the Polish side of the border in recent months and an unknown number on the Belarusian side.
Poland has been accused of pushing people back across the border into Belarus, contrary to international rules of asylum.
The migrants are mainly young men – but there are also women and children. They are camping in tents just inside Belarus, trapped between Polish guards on one side and Belarusian guards on the other.
Belarusian authorities say they are supplying them with food and heating.
“We have no place to sit, it’s too cold,” one of the migrants told the BBC’s Steve Rosenberg.
“We collect wood from trees to make heat. Still we hope, we never give up.