Friday's poll in the Yedioth Ahronoth daily found that 53% of those surveyed believe Sharon should resign as a result of the scandals, while 43% wanted him to stay on.
The former general has vowed to weather the storms kicked up by allegations of corruption and misconduct that he denies. But much will depend on whether prosecutors now weighing criminal charges against Sharon decide to indict him.
Already under investigation in two separate corruption scandals, Sharon came under another cloud this week when the Maariv newspaper alleged misconduct in a prisoner swap with Hizb Allah.
It reported that Sharon once had a business relationship with the father-in-law of Elhanan Tannenbaum, an Israeli businessman freed by the Lebanese group on 29 January in exchange for about 400 Arab prisoners.
The report suggested those links, dating to the 1970s, influenced Sharon's decision to push for the prisoner deal. Sharon denied any ulterior motive in winning freedom for Tannenbaum, who is under investigation for possible illegal business dealings at the time he was abducted in 2000.
The Yedioth Ahronoth poll also found that 57% of the Israeli public believes Sharon is not a trustworthy leader, up from 41% last summer.
The poll surveyed 501 Israelis and has a margin of error of 4.5%