Yamina Chairman Naftali Bennett will meet with both Prime Minister Benjamin Netanyahu and Opposition Leader Yair Lapid over the weekend.
The meetings with the right-wing lawmaker follow a week's silence during which he refused to identify the candidate he will recommend to President Rivlin to form the next government, insisting he will only do "what is right for the people of Israel."
Against the backdrop of Bennett's meetings with the Likud and Yesh Atid chiefs on Friday and Saturday evening, respectively, the right was bringing pressure to bear on Bennett not to join a government without Netanyahu, including the premier's Likud Party using various right-wing organizations in an attempt to cajole him.
Prime Minister Netanyahu on Wednesday called on Bennett and New Hope Gideon Saar to join his coalition as he struggles to muster up enough Knesset seats to form a government.
Israelis went to the polls for the fourth time in two years on March 23, but no clear victor emerged following the national vote.
Lapid, who has emerged as the leader of the anti-Netanyahu camp, has also called on Saar and Bennet to join the centrist bloc. Saar's New Hope and Bennett's Yamina parties have emerged as kingmakers as they hold enough seats to tip the scale for either side.
In a brief press conference, Netanyahu urged his former allies-turned-rivals to "come home" by uniting under him.
"It's no secret that we've had disagreements over the years, but we have known how to overcome them and work together," he said. "Let's form a stable right-wing government that will last for years. Come back home to your natural place - the Right."
He also said that any government that may be formed under Lapid would not last long.
"Any other government that is formed will be an unstable left-wing government, set up in clear opposition to the vast majority that voted for Likud and right-wing parties. A government with extreme opposition can destroy all our achievements - and it will fall very quickly," he added.
"The public gave the right-wing parties a clear majority with 65 seats. This makes it possible to immediately form a stable right-wing government. The challenges we face require a government - we must stabilize the economy, curb the Iranian nuclear program, strengthen settlements, sign peace agreements and bring more vaccines to our children so that we do not go back to lockdowns."
Saar brushed off the prime minister's calls, saying he will fulfill his election promise not to join a Netanyahu-led government.
Saar added that Netanyahu addressed him "on the very day that he and his people once again began spreading perverted and false conspiracy theories against myself as well as President Reuven Rivlin. The continued tenure of Netanyahu, who prefers his personal interests over the state's, harms Israel."
Bennett was less categorical in his reply, saying what concerns him is the people and "not appointments". Bennett said he "will continue to make every effort to establish a good and stable government that will pull Israel from the chaos."