South Korea's Kospi hits fresh record as hopes of U.S.-Iran breakthrough rise

South Korea's Kospi hit a fresh record Tuesday as trading resumed after a public holiday, with investor sentiment supported by hopes for a breakthrough in the U.S.-Iran peace talks.

President Donald Trump said Monday negotiations with Iran were "proceeding nicely," though he warned that the U.S. could resume attacks if the talks failed.

Oil prices were mixed after Trump's comments. West Texas Intermediate futures for July fell 5.37% to $91.41 per barrel as of 8:07 p.m. ET. International benchmark Brent futures for July rose about 1.75% to $97.82 per barrel.

In a sign of how tenuous the negotiations were, the U.S. Central Command said that it conducted "self-defense strikes" targeting Iranian missile launch sites and boats attempting to lay mines in the south of the Islamic Republic.

As the conflict wore on, Tehran appeared to be "blinking" over the Strait of Hormuz, ex-CIA director David Petraeus told CNBC's Lisa Kim at the UBS Asian Investment Conference on Monday.

South Korea's Kospi rose to a fresh high of 8,094.90 in early trade, while the small-cap Kosdaq added 2.12%

Japan's Nikkei 225 slipped 0.18% amid some profit taking, while the Topix declined 0.36%. The Nikkei 225 breached 65,000 for the first time Monday in holiday-thinned Asia trading.

Australia's S&P/ASX 200 fell 0.17% at the open.

Hong Kong's Hang Seng index futures were at 25,430, lower than the index's last close of 25,606.03 on Friday.

S&P 500 futures gained 0.78%, and Nasdaq-100 futures advanced 1.14%. Dow Jones Industrial Average futures popped 371 points, or 0.73%.

U.S. stock markets were closed Monday due to the Memorial Day holiday.

— CNBC's Fred Imbert and Lim Hui Jie contributed to this report.