Defence Minister Avigdor Lieberman also provided the government with some breathing space by announcing he did not intend to quit for now — even though he opposes the legislation upon which the compromise deal is based.
While the coalition crisis building in recent days was still not definitively resolved, Monday's moves could lessen the chances for elections for the time being.
Netanyahu has repeatedly said he wants the coalition to last its entire term, which ends in November 2019, but some coalition members suspect him of allowing a "fake" crisis to worsen to give him the option of forcing elections.
Polls suggest Netanyahu could remain prime minister in fresh elections despite the corruption allegations.
Some Israeli political commentators accuse Lieberman and Netanyahu of working together to preserve the option of fresh polls for the prime minister. Both men deny it.
"If there are elections, we shall take part and we shall win," Netanyahu told parliament on Monday evening. "But we are not there."
Making peace with Israel
Though Netanyahu's legal woes hang over the discussions, the dispute within the coalition centres on a separate issue entirely: legislation to exempt young ultra-Orthodox Jewish men from military conscription.
Lieberman opposes the legislation and wants to see the ultra-Orthodox serve in the military like their secular counterparts.
Ultra-Orthodox political parties have said they would not support a national budget for 2019 unless the bill is approved.
A compromise in the works would at least postpone the crisis, if not outright resolve it.
In a meeting late Sunday, ultra-Orthodox factions told Netanyahu they would agree to support the budget if the military conscription bill passed a ministerial committee and an initial parliamentary reading, postponing a final vote until the summer session.
On Monday, the ministerial committee gave approval to the bill.
Late in the evening the parliamentary finance committee approved the budget draft for the second and third readings needed to pass it into law, with committee chairman Moshe Gafni, of the ultra-Orthodox United Torah Judaism party voting in its favour, the parliamentary website said.
The budget was expected to put to its final vote on Tuesday or Wednesday, it added.
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Lieberman later told lawmakers from his Yisrael Beitenu party that he would continue to oppose the legislation, but would remain in the government for now and could resign later.
"We shall not volunteer to leave the government," Lieberman said.
"For as long as it has not passed its third (and final) reading, we shall fight from inside."
He added that "there is no more interesting post than the post of defence minister. I do not know of anyone who would volunteer to leave the job."
Netanyahu separately called on all coalition partners — "first and foremost Defence Minister Avigdor Lieberman"— to remain in the government.
Should Lieberman and his party quit, Netanyahu's coalition could in theory continue with a one-seat majority in parliament.
Netanyahu however says that he does not want to do so, calling it unsustainable.
The 68-year-old premier could soon face charges in at least two separate corruption affairs, while investigations are continuing into two others.
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Three of his former associates have signed state witness deals with police.
Police recommended his indictment for bribery in the first two cases in February and the attorney general is considering how to proceed, a process expected to take months.
In one case, Netanyahu and his family are accused of accepting expensive gifts from wealthy supporters in exchange for financial benefits or favours.
The other alleges he sought a secret deal with the publisher of a top-selling newspaper for favourable coverage.
Netanyahu has been prime minister for a total of 12 years, from 1996-1999 and again since 2009.
He is not required to step down if indicted — only after he is convicted with all appeals exhausted — though political pressure would surely mount.
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