Lieberman resigned from his post as defence minister earlier this week in protest of the agreed upon ceasefire between Israel and the Islamist group Hamas in the Gaza Strip.
Israel will pick a date to hold early elections on Sunday following a tumultuous week for the nation.
On Friday, Lieberman toured southern Israel and accused Netanyahu of being soft on terrorism.
The fishermen's union said the 20-year-old man was working on the beach near the land maritime fence separating Gaza from Israel when he was shot in the stomach.
After Friday's meeting, the source, who is close to Bennett, told CNN: "There was a need to go to elections as soon as possible, with no possibility of continuing the current government".
"It's impossible that after Hamas launches 500 rockets at the Israeli border communities. the heads of Hamas are actually getting immunity from the Israeli cabinet", he told reporters.
Number of missing grows to more than 1,000 in California’s massive wildfire
It has burned through 98,000 acres since it started on November 8, and is now 82 percent contained, though three people have died. As of Saturday, NBC reports that the fire is the deadliest in the United States in a century, with a confirmed death toll of 71.
Finance Minister Moshe Kahlon and Interior Minister Aryeh Deri both spoke to Netanyahu personally according to Haaretz, urging him to call early elections for the sake of the country's stability.
Representatives from Netanyahu's office said he would try to preserve his administration, but it looked unlikely after he lost Bennett's support.
The departure of Lieberman and his Israel Beitenu party had left the coalition with a one-seat majority in the 120-member parliament. Bennett's decision to withdraw the Jewish Home party would take away eight seats away from the prime minister's coalition, thus making elections inevitable.
Government figures would agree on a date on Sunday, the source added.
However, a survey conducted by Hadashot TV on Wednesday night revealed that Netanyahu's Likud party would win by a landslide if elections were to be held now.
But the crisis may pose a new challenge for the Israeli leader, since it appears he is being led to call early elections, instead of calling them on his own.