European shares pushed higher on Thursday, with investors betting Greece will be able to pull off a key bond swap as it strives to avoid a chaotic default, and turning more optimistic that companies with global exposure will do well.

Markets, however, were likely to remain volatile ahead of the evening deadline on the Greek swap offer to private creditors, as well as Friday's keenly-watched U.S. jobs report.

This has been the game of anticipation. I think it's likely to remain volatile but overall ... we would be expecting more upside potential moving into next week, Luca Solca, global head of European research at CA Cheuvreux, said.

Another 10-20 percent, I wouldn't be surprised at that ... I think the rally would take shape expanding into higher beta names.

The Euro STOXX 50 index of euro zone bluechips rose 0.3 percent to 2,453.19 points in early trade.

The broader FTSE Eurofirst 300 added 0.5 percent to 1,063.87. High-beta banks, which are most directly linked to Greece's debt worries through their sovereign debt holdings, added 1 percent.

(Reporting By Toni Vorobyova)