* FTSEurofirst 300 down 0.5 pct, Euro STOXX 50 down 0.2 pct
* Greece hours away from default, deal on Tuesday seen unlikely
* Luxury sector hit by BAML downgrade
By Alistair Smout and Atul Prakash
LONDON, June 30 (Reuters) - European shares fell again on Tuesday after a German government official said it was too late to consider an extension of Greece's bailout programme and Greece's finance minister said it will not pay a debt installment to the IMF due on Tuesday.
The pan-European FTSEurofirst 300 index and the euro zone's blue-chip Euro STOXX 50 index were down 0.5 percent and 0.2 percent respectively by 1126 GMT following the comments.
Greek Finance Minister Yanis Varoufakis said the country would not pay a 1.6 billion euro debt installment to the International Monetary Fund, 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," Peter Dixon, equity strategist at Commerzbank, said.
European stocks, which had opened lower after Monday's slump, had recovered in mid-morning trading after a report saying Greek Prime Minister Alexis Tsipras was considering a last-minute bailout proposal by the European Commission.
The Greek daily Kathimerini said the prime minister's office had told Brussels it was evaluating Monday's new proposal from European Commission president Jean-Claude Juncker that included debt relief in October.
"Tsipras might be facing a lot of pressure and thinking of a compromise as the outcome of the referendum is not going to help him either way," Koen De Leus, senior economist at KBC, said.
"If there is a 'yes' vote, Greece might see a new government and if the voters say 'no', then the country would head towards an economic chaos."
With Greece hours away from defaulting on a 1.6 billion euro loan from the International Monetary Fund, tens of thousands of Greeks rallied on Monday to back their left-wing government in a clash with foreign lenders that has forced a shutdown of its banking system.
Juncker said if Greeks rejected an offer from creditors to save them from bankruptcy in Sunday's referendum, it would be taken as a signal they wanted to quit the euro.
Euro zone stocks suffered their biggest one-day fall since 2011 on Monday, with southern European banks in particular getting pummelled after Greece shut its banks and imposed capital controls. The Athens stock market and main stock index are set to remain shut on Tuesday.
Christian Stocker, strategist at UniCredit in Munich, said a further decline in shares of some sectors such as personal household goods, food and beverages and healthcare could become a nice buying opportunity as the companies were dependent on consumer demand and should benefit from a pickup in the economy.
"But we think it's a bit early to buy these sectors as there are so many uncertainties. A further drop of about 5 percent in shares of these sectors would be a good level to enter."
Underperforming the market was the luxury sector, with Christian Dior down 3.4 percent.
LVMH and Tod's fell 2.1 percent and 1.6 percent respectively after all three firms were downgraded to "underperform" from "neutral" by Bank of America Merrill Lynch as they lowered earnings estimates for the luxury sector. (Editing by Mark Heinrich)