Barak, who initially rejected Netanyahu's offer, has asked his party to consider joining a Likud-led coalition, saying it was in the "superior interests of the state" to counter the effect of the rightist party.
The 1,460-strong Labour congress is due to decide on Tuesday whether they will join the new government.
"The battle will be decided by a very few votes," the Maariv daily said.
According to Israeli news reports, Netanyahu has offered Barak to stay on as the defence minister.
But Yuli Tamir, a Labour MP and outgoing education minister, claimed Barak was leading the party to its doom, saying a government led by Netanyahu would block any effort to relaunch the Israeli-Palestinian peace process.
Deal with Hamas
It is believed that the extra time would also give Ehud Olmert, the outgoing prime minister, more time to seek a prisoner-swap deal with Hamas to secure the release of Israeli soldier Gilad Shalit, who was captured by Palestinian fighters in Gaza in June, 2006.
The latest round of Egyptian-mediated negotiations over the prisoner swap collapsed on Monday following disagreements on the list of Palestinians to be released by Israel.
Netanyahu was asked on February 20 to form a government after February 10 elections. Initially he was given a 28-day deadline but is legally entitled to a two-week extension.
Although the rival Kadima party won 28 seats in the February elections, one more than Netanyahu's Likud, the former premier was tasked with forming the next government because he is believed to have a better chance of forging a majority in the 120-seat parliament.