Concern is growing over the cabinet choices of former Israeli prime minister Benjamin Netanyahu following his election victory last week.
Ultra far-right lawmaker Itamar Ben-Gvir is almost certain to claim a top ministerial role.
Israeli media is reporting that Aryeh Deri, who held the post of interior minister three times and was previously jailed for bribery, is expected to head either the finance ministry or the interior ministry.
Mr Deri leads the ultra-orthodox Shas party, the third-largest in Mr Netanyahu's right-wing bloc which was catapulted to power last week in a surprise comeback for the former leader, who has been embroiled in a corruption trial.
Mr Deri could be given the finance ministry, amid claims Mr Netanyahu wants to avoid giving the role to Religious Zionism chief Bezalel Smotrich, Ynet and Channel 12 have reported.
While Mr Netanyahu has yet to be formally asked to assemble a government, talks have begun to assign ministerial positions among Likud and its right-wing, religious allies.
Mr Netanyahu has offered Mr Deri his pick of ministerial positions, Channel 12 reported. Mr Deri is said to be asking for his former position of interior minister.
The Shas chief spent two years in prison in 2000 for taking bribes during his stint as Israel's youngest ever interior minister in the 1980s. He resigned from the Knesset in January as part of a plea deal in which he was handed a 12-month suspended prison sentence and a fine of 180,000 shekels for tax offences.
He was initially suspected of bribery but was finally charged with failure to report his income and other tax offences when selling Jerusalem apartments to his brother Shlomo.
In June, Shlomo was given nine months community service for multiple tax evasion charges, one of which included aiding Mr Deri by claiming income from a US investment fund as his own.
Mr Deri has held several government positions, including minister for development of the Negev and the Galilee.
He is not the only potential minister with a murky past.
Mr Ben-Gvir, widely slated to be the next minister of public security, has come under closer scrutiny in recent weeks for incendiary comments about Israel's large Arab minority.
He was convicted in 2007 of racist incitement and support for a group on both the Israeli and US terrorism blacklists.
He is also known for his support of Baruch Goldstein, who committed the 1994 Hebron mosque massacre, and his calls for Arabs to be deported from Israel.
The US is reportedly considering a “no-contact” policy with the Otzma Yehudit leader, and Jordan has allegedly warned ties will weaken if any change is made to the status of Al Aqsa Mosque, where Mr Ben-Gvir regularly visits.
“Bibi [Netanyahu] does not want Ben-Gvir and Deri to lead the dance,” Shlomo Fischer of the Jerusalem-based Jewish People Policy Institute told AFP.
“He is very careful. He does not want to lose his international legitimacy … I think he could try to widen his coalition to minimise their influence.”
