* Euro STOXX 50 extends Monday's slump, down 0.3 pct
* Greece just hours away from debt default
* Luxury sector hit by BofAMerrillLynch downgrade
By Atul Prakash
LONDON, June 30 (Reuters) - The euro zone's top share index steadied on Tuesday, a day after posting its biggest one-day percentage drop since 2011, as the European Union made a last-ditch bid to salvage a Greek bailout deal.
With billions of euros in locked-up bailout funds due to expire at midnight, the European Commission urged Greece to accept the proposed deal, while holding out hopes that some tweaks could still be possible.
Greece submitted a new two-year aid proposal, while German Finance Minister Wolfgang Schaeuble told lawmakers Greece would not have to leave the euro zone if Greeks voted against the bailout package in a referendum on Sunday.
"There have been concerns that Greeks have shot themselves in the foot, closed the door and cut themselves off from funding, leading to an exit from the euro zone. But Schaeuble made some conciliatory remarks, giving hopes that the door is still open to find a solution," B Capital Wealth Management managing director, Lorne Baring, said.
"For any investors with risk tolerance, this is an opportunity to buy European equities. The economic picture is broadly improving and has got value, supported by the European Central Bank's quantitative easing programme."
The euro zone's blue-chip Euro STOXX 50 index was down just 0.3 percent by 1348 GMT after Monday's slump of 4.2 percent. Both Germany's DAX and the pan-European FTSEurofirst 300 indexes fell 0.5 percent.
In choppy trading, stocks opened lower and then turned positive after a report saying Greek Prime Minister Alexis Tsipras was considering a last-minute bailout proposal by the European Commission.
The Athens stock market was shut on Tuesday.
Stoking investor jitters, Greek Finance Minister Yanis Varoufakis said the country would not pay a 1.6 billion euro debt installment to the IMF, but Athens still holds out hope of a last-minute deal on an aid package.
"All these mixed messages are making investors more nervous and there is potential for a lot more wobbles to go in the next few days. The possibility of a deal is slim, but it's not zero," Commerzbank equity strategist, Peter Dixon, said.
Underperforming the market was the luxury sector, with Christian Dior down 3.6 percent.
LVMH and Tod's fell 2.7 percent and 2.1 percent respectively after all three firms were downgraded to "underperform" from "neutral" by Bank of America Merrill Lynch as it lowered earnings estimates for the luxury sector. (Additional reporting by Alistair Smout; Editing by Louise Ireland)