Brazil's new leader made the comments last Thursday as he met with evangelical leaders and was talking about a visit he made early this month to the Yad Vashem Holocaust museum in Jerusalem.
"We can forgive but we cannot forget. And that is my statement. Those who forget the past are condemned to having no future," Bolsonaro had said.
Israeli President Reuven Rivlin took to Twitter in response, although without mentioning the Brazilian leader by name.
"We will always oppose those who deny the truth or those who wish to expunge our memory -- not individuals or groups, not party leaders or prime ministers. We will never forgive and never forget," Rivlin wrote.
In a letter sent to Israeli authorities over the weekend, Bolsonaro insisted that he had been misunderstood. He referred again to his visit to Israel.
"In the guest book at the Holocaust Museum, I wrote 'Those who forget the past are condemned to having no future.' So any other interpretation only interests those who want to distance me from my Jewish friends," he wrote in the letter, which was posted on social media by the Israeli ambassador to Brazil, Yossi Shelley.
Bolsonaro raised eyebrows in other ways during his visit to Israel, saying for example that Nazism was a leftist political movement.
In his new letter, he sought to clarify his concept of forgiveness.
"Forgiveness is something personal, never in an historic context such as the case of the Holocaust, in which millions of innocent people were murdered in a cruel genocide," Bolsonaro wrote.
Shelley defended Bolsonaro, saying his letter left no doubt that he believed the Holocaust was "the biggest genocide in history." Shelley called the president a friend of the Israeli government and people.
Bolsonaro has sought close ties with Israel since coming to power in January. He has pledged to follow the US lead in moving his country's embassy to Jerusalem. And Prime Minister Benjamin Netanyahu was one of few foreign heads of state or government that attended Bolsonaro's inauguration.
At home, Bolsonaro is a former military man who has praised the country's time as a dictatorship from 1964 to 1985 and made remarks that insulted blacks, gays and women.