Brent crude <LCOc1> futures rose $1.88 to settle at $65.49 a barrel, a 2.96 percent gain. Brent traded between $63.69 and $65.63 during the session.
West Texas Intermediate (WTI) crude <CLc1> futures rose $1.92 to settle at $62.04 a barrel, a 3.19 percent gain. U.S. crude traded between $60.14 and $62.14.
Oil prices extended gains in post-settlement trade, with both benchmarks up more than $2.
Both Brent and U.S. crude notched weekly percentage gains after weekly percentage losses last week.
Trump said he was prepared to meet Kim in what would be the first face-to-face encounter between leaders from the two countries and could mark a breakthrough in a standoff over the North's nuclear weapons. Kim also has committed to suspending further nuclear or missile tests.
Wall Street jumped after U.S. payrolls data showed strong job additions in February. The S&P 500 index <.SPX> was last up about 1.5 percent, while the Nasdaq Composite index <.IXIC> hit a record high on the day, further supporting oil prices. Crude futures and stock indices have recently moved in tandem.
The jobs report "speaks to strong, underlying economic conditions, and growth, which includes increased energy demand," said John Kilduff, partner at investment manager Again Capital in New York.
Also bullish for oil prices, Libya's 70,000 barrels per day El Feel oilfield stayed shut despite the Petroleum Facilities Guard saying it had reached a deal to reopen it, according to a field engineer and local mediator.
Hedge funds and money managers cut their bullish wagers on U.S. crude for the first time in three weeks, data showed on Friday.
The speculator group cut its combined futures and options position in New York and London by 17,166 contracts to 478,531 in the week to March 6, the U.S. Commodity Futures Trading Commission (CFTC) said.
The reduction came as gross short positions on the New York Mercantile Exchange <3067651MSHT> climbed to the highest in nearly a month.
U.S. oil drillers cut four oil rigs in the week to March 9, bringing the total count down to 796, General Electric Co's Baker Hughes energy services firm said in its closely followed report on Friday.
"Sometimes you're going to have week-to-week variations which are going to be negative, but I wouldn't take too much from any single week," said Stewart Glickman, an energy equity analyst at CFRA Research in New York.
"The bigger thing is that the three-month average is definitely moving higher."
(Additional reporting by Shadia Nasralla and Libby George in London, and Henning Gloystein in Singapore; Editing by Cynthia Osterman and David Gregorio)
By Stephanie Kelly