The dispute began on Monday, just days before a planned visit by US President Donald Trump, with Israel angrily demanding an explanation from the White House over the comments.
Since Trump took office last month, Israel has announced plans to build over 6,000 settler homes in the West Bank and east Jerusalem - occupied territories claimed by the Palestinians.
Israel will be the second stop on Trump's first foreign trip, following Saudi Arabia.
Trump is reportedly planning to visit the wall in Jerusalem's Old City and Israel's Channel Two reported that Israeli officials offering to help plan the event were told by American counterparts it was not their remit.
Rivlin also mentioned that Israel was looking forward to hosting President Donald Trump next week, expressing "a sign of the unbreakable bond between Israel and the US".
US presidents of both parties have repeatedly waived a USA law requiring the embassy be moved to Jerusalem. His inner core of top advisors was populated with enthusiastic proponents of expanding Israeli settlements in the West Bank and, in a highly-contested move, the president appointed a very partisan David Friedman as ambassador to Israel.
The White House distanced itself from the alleged comments, saying in statements to US media that they did not reflect the views of the administration.
The new United States ambassador to Israel, David Friedman, visited the Western Wall on Monday, in a departure from diplomatic protocol. "When President Trump arrives in Israel, I don't think there will be any doubt over who has title over the Kotel", he said. Despite the Kotel's significance in Judaism and Israel's control over a united Jerusalem, the worldwide community-including the US -does not officially recognize the Western Wall as part of Israel.
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Friedman, who owns an apartment in Jerusalem, is expected to live in the USA ambassador's official residence in the Tel Aviv suburb of Herziliya.
The Western Wall - also known as the Wailing Wall - was occupied along with rest of east Jerusalem in the 1967 war in a move not recognised by the global community.
Israeli ultra-Orthodox Jewish men participate in the Cohanim Priestly caste blessing during the Jewish holiday of Passover, in front of the Western Wall, the holiest site where Jews can pray in Jerusalem's old city, Monday, April 25, 2016.
Speaking at a pre-election rally in October in support of Trump on Jerusalem's Mount Zion, Friedman said Trump would fire USA state department "lifers" who refused to move the embassy.
General H.R. McMaster refused to answer if the Western Wall belonged to Israel.
He has also advocated moving the embassy to Jerusalem.
Addressing the press following the ceremony, Rivlin said that Israel looks forward to welcoming President Trump next week. "Those are unworkable suggestions that are never going to be agreed upon by the State of Israel".