Prime Minister Benjamin Netanyahu offered a specific proposal on Tuesady to Habayit Hayehudi. Sources in the Prime Minister's Office described Netanyahu's proposal to Naftali Bennett's party as "very respectable."

The proposal includes a series of high-level government appointments for Habayit Hayehudi and the opportunity to exercise influence in setting the new government's priorities.

Under the proposal, Habayit Hayehudi would receive the education portfolio and a top-level economic portfolio, though not the finance portfolio. The party would also get the deputy defense minister position, which would include broad authority over construction in Judea and Samaria. Likud-Beytenu is also willing to grant Bennett's party the chairmanship of the Knesset Finance Committee.

Likud-Beytenu officials told their Habayit Hayehudi counterparts that the offer would be on the table for only 48 hours, after which Likud-Beytenu would turn to other parties.

Habayit Hayehudi officials said on Tuesday that they had not heard of the proposal, calling it spin intended to drive a wedge between Habayit Hayehudi and Yesh Atid. However, Likud-Beytenu officials confirmed that the proposal was real and had been presented to Bennett.

Bennett on Tuesday again signalled that his alliance with Yesh Atid leader Yair Lapid remains strong. Speaking in Jerusalem to the Conference of Presidents of Major American Jewish Organizations, Bennett said that he and Lapid have substantial common ground on the economy and the distribution of the national service burden.

While addressing the same group on Tuesday, Lapid actually highlighted some of the differences between him and Bennett. Among other things, Lapid criticized Israel for not honoring its part of the "Road Map" for peace regarding settlement construction.

Lapid did state, however, that he is opposed to any future division of Jerusalem in a potential peace agreement with the Palestinians. Lapid said that the government of former Prime Minister Ehud Olmert went too far, too soon, when it discussed Jerusalem and the right of return with the Palestinians.

Meanwhile, Shas is continuing its efforts to join the coalition. A day after meeting with Bennett, Shas leader Eli Yishai met with Uri Ariel, No. 2 on the Habayit Hayehudi list. Yishai and Ariel discussed a number of political issues, including various proposals for haredi (ultra-Orthodox) enlistment.

The haredi parties are still unsure what the chances are that they will be part of the government.

"It all depends on the prime minister," a Shas official told Israel Hayom. "It appears that the alliance between Bennett and Lapid is strong, which could leave Netanyahu with no choice."