Oren Hazan was not supposed to approach Trump during the welcoming ceremony at Tel Aviv airport, but managed to "get in" to the line of dignitaries, a foreign ministry spokesman said without elaborating.
As Trump passed, the never-shy Hazan whipped out his cellphone and suggested a selfie to Trump, television footage showed.
After a bewildered look Trump agreed, although Netanyahu tried to prevent it by gently pushing Hazan's arm.
Within a few hours, the selfie had 17,000 "likes" on Hazan's Facebook page.
Oren Hazan takes a selfie with U.S. President Donald Trump at welcoming ceremony in Tel Aviv. Source: Israeli Government Press Office
But it also attracted the anger of some internet users, especially in the right-wing Likud party of which Hazan and Netanyahu are both members.
"We have to get Oren Hazan out of parliament, and the sooner the better," said Benjamin Lachkar, a member of Likud's central committee, on Facebook.
Other Internet users hijacked the photo, adding speech bubbles joking about the US president's pride of having a selfie with Hazan.
Another picture shows Hazan smiling while posing next to Christ on the cross.
Hazan was suspended in 2015 from his position as deputy speaker of Israel's parliament after a televised report accusing him of involvement in pimping and drugs.
He was also suspended for one month from any parliamentary activity in December 2015 after "insulting" behaviour towards a colleague with a disability.
Asked about his exchange with the US president, Hazan told media that he was often referred to as the "Israeli Trump".
"It was for me - and I think for him also - a historic moment," he added.
