Rival Gantz, 59, earlier also claimed victory, citing preliminary exit polls that showed his party had won more seats than Likud.
"The skies may look overcast. but they can not hide the sun of hope that we have brought to the Israeli people and society", Gantz, 59, wrote in an open letter.
The big losers of this election appear to be the left parties, including Labor, which is projected to lose 13 seats, Meretz, projected to lose one, and the United Arab List-Balad, which shaved 3 seats off its Knesset presence.
He thanked the US President for his "tremendous support for Israel". "I will begin forming a right-wing government with our natural partner this very night". No single Israeli party has ever won enough seats to form a government, and all governments have been coalitions of several parties.
Gantz said he favoured a "globally backed peace agreement" with Israel holding on to the large West Bank settlement blocs, adding that he opposed unilateral moves.
The results do not include the votes of soldiers, prisoners and diplomats, which are expected to be counted by Thursday afternoon and could affect the numbers.
A close result in the election would put smaller parties in a powerful position, turning marginal political figures into kingmakers.
Despite that, Mr Netanyahu gained four seats compared to his outgoing coalition government, according to a spreadsheet published by the Central Elections Committee.
"We respect the decision of the people", Gantz told journalists, acknowledging his centrist Blue and White alliance had failed to unseat Netanyahu in Tuesday's poll.
The race between Gantz's and Netanyahu's coalitions attracted worldwide attention. Therefore Bibi may still be the one asked by Israel's President Rivlin to form a government (BTW asking a party leader to form a government is the only real power held by the Israeli President).
Though both parties were well short of a majority in the 120-seat parliament, the polls showed Likud and its religious and nationalist allies controlling a solid majority.
USA officials say they have briefed regional leaders, though the Palestinian Authority says it has been boycotting diplomatic contacts with Washington to protest President Trump's recognition of Jerusalem as Israel's capital. Lapid is a nemesis of the ultra-Orthodox; they wouldn't recommend him as prime minister.
In a rare turn during the race towards the Israeli-Palestinian conflict, Mr Netanyahu further alarmed Palestinians by promising to annex Jewish settlements in the occupied West Bank if re-elected. Along with two other former military chiefs on his ticket, Gantz has attacked Netanyahu for failing to halt rocket fire from the Hamas-ruled Gaza Strip. But as the night went on, there were growing signs that Netanyahu's Likud was pulling ahead.
How do Israeli elections work?
USA officials have so far only revealed vague outlines of what the plan might propose, but Secretary of State Mike Pompeo suggested it could break abruptly with precedent.