"Barghouti, if you are going to break your (hunger) strike, isn't pizza the better choice?" a screenshot of the post said, with a pizza box seemingly photoshopped into the picture.
A flood of tweets in Hebrew have suggested those who support the cause "Send a chocolate bar and Pizza Hut pizza to MK Odeh" to help him with his hunger strike.
Voting for around 300 municipal councils opened at 7:00 am (0400 GMT) on Saturday at dozens of schools across the West Bank, which has been occupied by Israel for half a century.
Some 850,000 Palestinians have been incarcerated since the Israel's occupation of their territories 50 years ago, Palestinian leaders say.
According to Palestinian figures, Israel is now holding more than 6,500 Palestinians - including scores of women and children - at 24 detention centers across the country.
But for Israel, Barghouti is a terrorist.
"I hope in the near future [Odeh] will go on hunger strike between the walls of prison, for supporting and inciting terrorism", Ilatov added.
"When it comes to the prisoners issue there is no time - there's no time for anybody to wait while we are talking", he said.
Barghouti, a senior member of Palestinian president Mahmoud Abbas's Fatah party and a highly popular figure among Palestinians, is serving five life sentences over his role in the second Palestinian intifada, or uprising. Several hundred are being held without trial in so-called administrative detention.
Bargouthi's wife, Fadwa, also rejected the allegations in a press conference in Ramallah, stressing that Israel was "resorting to despicable acts" and that "the prisoners are familiar with Israel's lies and games and the video they released signals the beginning of its fall", and will only make the prisoners more determined.
Associated Press writer Karin Laub in Ramallah, West Bank, contributed to this report.
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