Gold prices also climbed, boosted by a weaker U.S. dollar after U.S. President Donald Trump announced the disbanding of two high-profile business advisory councils, and minutes from July's meeting of the Federal Open Markets Committee showed that policymakers had grown increasingly anxious about weak U.S. inflation readings. Last week's fear was prompted by U.S. President Donald Trump's warning North Korea would face "fire and fury" if it threatened the United States and North Korea's announcement it was considering plans to fire missiles at the U.S. island territory of Guam.
Still, with joint South Korean-US military exercises scheduled and North Korea celebrating "Liberation Day" Tuesday, volatility could return to markets, analysts said.
Japan's Nikkei stock index closed up 1.1 percent, boosted by the weaker yen, a day after skidding 1 percent to its lowest since early May. Sompo Holdings rose 2.2 percent, Mitsubishi UFJ Financial Group rose 1.7 percent and Mizuho Financial Group added 1.4 percent.
The dollar index was last up 0.45 percent at 93.848.
U.S. stocks rebounded on Monday, reversing some of the losses that were triggered last week by rising tensions between the U.S. and North Korea.
Oil prices steadied somewhat after falling more than 2.5 percent on Monday to its lowest in about three weeks on the strength of the dollar and reduced refining in China.
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PREMIER League bosses are to be given a say on whether there should be changes to the summer transfer window. Klopp added: "It makes sense that when the season starts, the planning is over".
The dollar rose broadly on Monday as traders unwound bearish bets against the US currency that have come in the wake of increasing tensions with North Korea and underwhelming inflation data. Apple, Microsoft and Facebook each rose about 1 percent in the first few minutes of trading.
After rising last week on investor concerns over geopolitical risks, both the Swiss franc and the yen have pulled back over the past couple of days as such worries waned.
The euro was down 0.4 percent at $1.1733, helping push the dollar index, which measures the greenback against a basket of currencies, up 0.4 percent. Bank of America climbed 2.3 percent and Wells Fargo gained 1.7 percent.
Gold, viewed as a safe haven for investors in troubled times, fell 0.6 percent to $1,274 an ounce.
The Standard & Poor's 500 rose 25 points, or 1 percent, to 2,466.
U.S. West Texas Intermediate crude futures settled down $1.23, or 2.52 percent at $47.59 a barrel.