The Judicial Selection Committee is expected to meet on Wednesday in order to vote on the four candidates.
In order to be selected as a Supreme Court judge, a majority of seven out of nine votes is required.
One of the candidates is the head of the Haifa District Court, Judge Yosef Elron, who is considered a conservative. Elron is supported by Finance Minister Moshe Kahlon (Kulanu) and Minister of Justice Ayelet Shaked (Bayit Yehudi), who is head of the selection committee.
Chief Justice Miriam Naor has claimed in recent months that several of her colleagues think Elron is "not the stuff of a Supreme Court judge." However, several judges disagree, expressing that they find him worthy of the post.
Kahlon convinced Naor to agree to Elron's appointment as a part of a "package deal" in which Naor will also get the nominations of the other three judges she wanted.
However, last week, the soon-to-be chief justice, Esther Hayut came out against Elron's appointment, bringing an end to the aforementioned deal.
If by Wednesday an agreement is not reached, the committee will disband without a vote.
The four new candidates are meant to replace Elyakim Rubenstein, Zvi Zylbertal, Salim Joubran and Chief Justice Miriam Naor.
If no agreement is reached, Shaked and Kahlon will not be able to prevent a bill that will allow for the appointment of a Supreme Court judge by a simple majority.