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.

In order to form the kind of “full-on right-wing government” he promised his voters, he will need the support of another small right-wing party that has been sitting on the fence. He will also need the far-right flank of his potential coalition to agree to rely on the support of a small Arab, Islamist party that has become a linchpin.

So far, Mr. Netanyahu’s partners on the far right have rejected that proposition. The other option is for Mr. Netanyahu to woo some defectors from the opposite camp.

“We will have to be creative,” said Danny Danon, a former Likud minister who served as Israel’s ambassador to the United Nations. “Everything will be on the table. People will have to be flexible,” he said, “including Netanyahu.”

Mr. Netanyahu has been charged with bribery, fraud and breach of trust. He has denied all wrongdoing.