JERUSALEM — Prime Minister Benjamin Netanyahu of Israel was tasked by the president on Tuesday with trying to form a new coalition government, offering a possible new path for him to stay in office even as he stands trial on charges of corruption.

A political survivor and Israel’s longest-serving prime minister, Mr. Netanyahu has spent the last 12 years in office. But after four inconclusive elections in two years, he and his allies have failed to win enough support to ensure a parliamentary majority that could end the country’s political deadlock.

Mr. Netanyahu now has 28 days to try to cobble together a coalition that could command a majority of 61 in the 120-seat Parliament, with the possibility of an additional 14-day extension. If he fails, the president, Reuven Rivlin, could task another candidate or refer the matter of choosing a candidate to Parliament.

In last month’s election, Mr. Netanyahu’s conservative Likud party emerged as the largest, with 30 seats. Together with his allies in the right-wing and religious camps, he has 52 seats. That falls short of a majority, but it is still more than any of his opponents managed to muster.