In a statement, Mr Netanyahu said he had worked "tirelessly" to establish a unity government with his chief rival, former military chief Benny Gantz, but been repeatedly rebuffed.
If he eventually failed to form a government when the 28 days deadline elapses, his rival Benny Gantz is expected to take over to try to achieve same.
Likud placed second in the September ballot with 32 seats in the 120-member parliament, behind 33 for Blue and White.
The bloc "is determined to form a liberal unity government under Benny Gantz, who the nation elected a month ago".
After Netanyahu's announcement on Monday, Rivlin said he would consult with various political parties to let them know that he intends to ask Gantz to form a government.
The office of President Reuven Rivlin said on Monday evening that Netanyahu had announced he would be "returning the mandate to form the government to the president because he was unable to do [so]".
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Though Gantz's Blue and White party won the most seats in the last election, the former Israeli army chief does not have a clear path to forming a parliamentary majority.
Blue and White party Leader Benny Gantz looks on during his party faction meeting at the Knesset, Israel's parliament, in Jerusalem, on October 3.
He and Gantz both call for a national unity government, but both will have to make serious compromises to get there and there is little sign that is likely yet.
He then opted to have parliament dissolve itself and send the country to a re-vote in September, rather than risk another politician getting the chance to lead negotiations.
In contrast to Mr Netanyahu, whose political career spans three decades, the 60-year-old Mr Gantz is a newcomer who only burst onto the scene over the last year.
For the time being, Likud has remained steadfastly behind its leader. He will have 28 days to entice potential allies.