he BBC's Adam Easton reports on the growing row over alleged anti-Semitic remarks made by the controversial head of a religious radio station in Poland.
Fr Rydzyk said tapes of his alleged comments have been doctored
In it, Fr Rydzyk, reportedly criticises Mr Kaczynski for his subservience to the Jewish lobby.
"You know that it's about giving $65bn," to the Jews, he allegedly said. "They will come to you and say 'give me your coat. Take off your pants. Give me your shoes'," the magazine reported.
The Simon Wiesenthal Center - an international Jewish human rights organisation - strongly rebuked the comments.
"This is outrageous, a [Nazi propaganda minister] Josef Goebbels in a collar," said Rabbi Marvin Hier of the Center.
Israel's ambassador to Poland, David Peleg, said Fr Rydzyk's comments were damaging Polish-Israeli relations.
"The tapes that have come to light are evidence that Rydzyk and his institutions are anti-Semitic and something must be done in the name of good Polish-Israeli relations," he said in an interview with Poland's Gazeta Wyborcza newspaper.
Fr Rydzyk, a priest from the Redemptorist order, has regularly courted controversy since he launched Radio Maryja in 1991.
With a blend of chat, music and religious services, the station styles itself as the "Catholic Voice in Your Home" and attracts many listeners in rural and poorer areas.
I... expected the Polish government and president would have reacted at once in a decisive way to Rydzyk's anti-Semitism
Israeli ambassador to Poland
Figures suggest Radio Maryja is steadily losing listeners with the station now attracting just over a million people or around two per cent of the daily audience.
But Fr Rydzyk's political influence has never been higher. In 2005, he urged his listeners to vote for the Law and Justice party, headed by Mr Kaczynski's twin brother, Jaroslaw.
The party won the elections and subsequently its members have become regular guests on both the radio and TV stations.
Last weekend, for the first time ever, a prime minister appeared at Radio Maryja's annual pilgrimage.
Addressing the crowd of around 150,000 people, Jaroslaw Kaczynski told them: "Here is Poland."
A day later, the magazine, Wprost, published its story.
Apart from the alleged anti-Semitic remarks, it also claimed Fr Rydzyk called President Kaczynski's wife a "witch" for her supposed support of limited abortion rights.
"You witch! I'll let you have it. If you want to kill people, do it to yourself first," he reportedly said.
Fr Rydzyk has suggested the tapes were doctored and called the story "fictitious".
The magazine says the authenticity of the tapes is beyond doubt.
Poland's Senate Speaker, Bogdan Borusewicz, has also been quoted as saying he believes the tapes to be genuine.
Despite the alleged slur, both Lech and Jaroslaw Kaczynski have refused to condemn the priest.
If the tape is verified as being Fr Rydzyk, then he should apologise, they said.
Many commentators here believe the twins may be reluctant to move against him.
The priest has the ability to carry perhaps more than a million votes to their side come new elections.
However, Mr Peleg is hopeful action will be taken.
"I also expected the Polish government and president would have reacted at once in a decisive way to Rydzyk's anti-Semitism. And although a few days have passed, I think this reaction will be forthcoming," he told Gazeta Wyborcza.