Israeli leaders have repeatedly spoken of improving ties with the Arab world and hinted at behind-the-scenes cooperation, particularly involving their efforts to limit their common enemy Iran.
Katz issued the call during an interview with Elaph, a news website run by a Saudi businessman, his spokesman Arye Shalicar said.
The call was not included in the final version of the interview published online. Shalicar could not comment on why, but said Katz had indeed made the comments.
"He said that he asks the king to invite (Prime Minister Benjamin) Netanyahu officially to Riyadh, and he asks MBS, Mohammed bin Salman, the son, to come and visit Israel," Shalicar said of Katz, who is also transportation minister.
Katz made the call because "he wants regional peace -- that's part of his three-layered program," his spokesman said, referring to security and economic cooperation leading to eventual peace.
He referred to Saudi Arabia as a leader in the Arab world.
Netanyahu has spoken repeatedly and with pride about growing rapprochement with "moderate Arab states" without naming them, although he is assumed to be referring to Saudi Arabia and other Gulf monarchies.
In November, Israeli military chief of staff Gadi Eisenkot said in an interview with the same website Katz spoke to that his country was prepared to cooperate with Saudi Arabia to face Iran's plans "to control the Middle East."
Arab countries have not made similar statements publicly and continue to criticise Israel over its 50-year occupation of Palestinian territory.
US President Donald Trump's recognition of Jerusalem as Israel's capital last week has drawn sharp criticism from Saudi Arabia.
King Salman said Wednesday that Palestinians have the right to Israeli-annexed east Jerusalem as their capital.
Only two Arab countries -- Jordan and Egypt -- have peace treaties with Israel.
In Katz's interview on Wednesday, he also issued a warning to Lebanon and Shiite group Hezbollah, which is based in the neighboring country.
Israel and Hezbollah fought a war in 2006 and often trade warnings.
Katz said if Hezbollah initiates a war against Israel, "this time all of Lebanon will be a target."
"What happened in 2006 would be a picnic compared to what could happen... I say Lebanon will return to the stone age," he threatened.