Israeli right-wingers photoshop president wearing Palestinian keffiyeh
Israeli soldier Elor Azaria was convicted of manslaughter in January for shooting dead the incapacitated Palestinian Abdul Fatah al-Sharif in Hebron last year.
The killing highlighted the deep divisions in Israel and led to an extraordinary rift between right-wing politicians - who wanted to see Azaria released - and top military brass, who harshly condemned his actions.
Rivlin on Sunday formally rejected a request to pardon Azaria, drawing the ire of Israel's notoriously vocal right-wing.
Israeli politicians criticised Rivlin's decision, while far-right Culture Minister Miri Regev went as far as saying that he had "abandoned Elor Azaria and harmed the pardoning institution".
The Israeli president's Facebook page quickly filled with messages of support but also scathing criticism. Some accused him of "fawning to appease your Arab and left-wing friends".
Others told him that he was "no longer my president".
Populist lawmaker Oren Hazan of Rivlin's Likud party called on the president to resign and said the pardoning authority should move from the presidency to the parliament.
Israeli police said they launched a probe after an image of Rivlin wearing a Palestinian scarf went viral.
The keffiyeh was made famous internationally after it became the trademark attire of late PLO leader Yasser Arafat and a symbol of resistance - as well as a cultural motif - among Palestinians.
The photoshopped picture features a smiling Rivlin with what appears to be the symbol for Hamas in the top left corner.
Text in Hebrew across the photo reads: "Reuven Rivlin a traitorous Jew-boy, may his name and memory be accursed".
The police statement did not elaborate on those being investigated nor the possible crime committed.
The image is reminiscent of similar depictions of slain Prime Minister Yitzhak Rabin which had appeared before his assassination by a right-wing Israeli extremist in 1995 following the Oslo Accords.
The image was meant to equate Rabin with Arafat and imply he was a traitor.
Opposition head Isaac Herzog of the Labor party published Rivlin's doctored image next to Rabin's.
"We all know how it starts, we all know how it ends," he wrote on Twitter.
Rivlin's office meanwhile postponed an olive harvesting event at the presidential residence in Jerusalem set for Monday morning, citing "inclement weather", despite clear skies.