Gantz is the first politician other than Prime Minister Benjamin Netanyahu to receive such a mandate since 2009.
Following deadlocked elections on September 17, Netanyahu had tried to form a coalition, but finally gave up on Monday - his second such failure this year.
At a press conference in Jerusalem, Rivlin called on political parties to make "concessions", while Gantz promised to "try to form a liberal union government".
He is expected to face difficulties in forming a majority coalition, despite expressing confidence he can reach a deal for a unity government.
He will have 28 days to try and if he too fails, Rivlin can ask parliament to agree on another candidate for prime minister.
If that also fails to produce a new government, Israel could face yet another election - its third in the space of a year.
"We must behave responsibly towards Israeli citizens and avoid new elections," Gantz said on Wednesday, adding that there would be room for "all elements of Israeli society" in his coalition.
Further negotiations
Gantz presents himself as a leader who can heal Israel's divisions, which he says Netanyahu has exacerbated.
A 60-year-old former paratrooper, Gantz had no previous political experience when he declared himself Netanyahu's electoral rival in December.
He was born on June 9, 1959, in Kfar Ahim, a southern Israeli village that his immigrant parents, both Holocaust survivors, helped to establish.
He joined the army in 1977, completing the tough selection course for paratroopers.
According to his official army biography, he was Israel's military attache to the United States from 2005 until 2009.
He was chief of staff from 2011 to 2015, when he retired, and has boasted in video clips of the number of Palestinian militants killed and targets destroyed under his command in the 2014 war with Gaza's Islamist Hamas rulers.
A security hawk, he is determined - like Netanyahu - to keep the Jordan Valley in the occupied West Bank under Israeli control and to maintain Israeli sovereignty over annexed Arab east Jerusalem.
The two are also in step on external threats, such as from archfoe Iran and its Lebanese ally Hezbollah.
Gantz has pledged to improve public services and show "zero tolerance" for corruption - a reference to graft allegations facing Netanyahu.
Negotiators from Gantz's Blue and White party and Netanyahu's Likud party will meet on Thursday, according to Likud.
Both the Likud and Blue and White say they want a big-tent coalition, but they are divided on how to achieve it.
The Likud has been seeking to negotiate based on a compromise set out by Rivlin that takes into account the possibility the premier will be indicted on corruption charges in the coming weeks.
It could see Netanyahu remain prime minister for now, but step aside at some point later as he fights the charges.
Gantz would take over as acting premier under such a scenario.
Blue and White says Gantz should be prime minister first under any rotation arrangement, since his party won the most seats, finishing with 33 compared to the Likud's 32 in the 120-seat parliament.
Rivlin promised on Wednesday night that he would do everything possible to avoid a third election.
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