Asian shares got the week off to a strong start and U.S. stock futures and the euro firmed on Monday, after Greece scrambled to avert defaulting on its debt with last-minute proposals aimed at appeasing its creditors.

Many investors were still cautious about Prime Minister Alexis Tsipras's proposals because it was not immediately clear how far they yielded to creditors' demands for additional spending cuts and tax hikes, nor whether creditors can stomach the offer.

The chief-of-staff to European Commission President Jean-Claude Juncker sent a tweet calling the latest proposal from Greece to resolve its debt crisis a "good basis for progress" in talks at an emergency euro zone summit meeting later on Monday.

"Today will supposedly be the last round of negotiations, so there could be a possible surprise in the form of an agreement with both parties seeing the writing on the wall," said Shinichiro Kadota, chief Japan forex strategist at Barclays in Tokyo.

"Still, it is unclear when and if an agreement will be reached, and the euro will remain sensitive to headlines over Greece."

MSCI's broadest index of Asia-Pacific shares outside Japan was up 0.5 percent, while Japan's Nikkei stock index added 0.8 percent.

U.S. stock futures tacked on 0.4 percent, while the euro also ticked up 0.2 percent to $1.1372, moving back toward a one-month high of $1.1440 hit on Thursday. The greenback was flat at 122.71 yen.

Athens is running out of cash to repay a 1.6 billion euro IMF loan due at the end of the month unless it secures new financing from international creditors.

Speculation is rife that, if no deal were reached on Monday, Greece may need to impose capital controls on Tuesday to avert a banking crisis as savers keep withdrawing funds from banks.

Bank of Greece Governor Yannis Stournaras met senior bankers on Friday and told them to brace for a "difficult day" on Tuesday if no deal is reached, two bankers at the meeting told Reuters.

In Asia, Chinese mainland markets were closed on Monday for a holiday, after posting their biggest weekly loss since 2008 last week.

Some strategists said such a correction was long overdue as Chinese shares had at one point marked gains of more than 40 percent this year. But others fret that falling stock prices could stoke concerns about a slowdown in the Chinese economy.

In commodities trading, copper futures rose about 0.2 percent $5,669.50 a ton, moving away from last week's three-month low hit on worries about demand in top metals consumer China and uncertainty in the Greek debt talks.

Crude oil futures dipped amid caution about Greece, and worries over high U.S. oil product. Brent crude fell about 0.3 percent to $62.86 a barrel, while U.S. crude futures slipped about 0.2 percent to $59.48.