BRUSSELS, Dec 13 (Reuters) - The European Union pressed the Israeli government on Monday to freeze settlement building, offered the Hamas-ruled Gaza Strip more aid and urged Israel to open Gaza's border crossings more fully to increase trade.

EU foreign ministers noted with regret Israel's failure to extend a moratorium on construction of Jewish settlements, and took a stand at odds with the decision by the United States to drop efforts to persuade Israel to freeze settlement building.

Our views on settlements, including in East Jerusalem, are clear: they are illegal under international law and an obstacle to peace, the ministers said in a statement after a meeting in Brussels.

Israel has hailed Washington's policy shift, but Palestinian officials have been highly critical, saying Israeli settlement building on land they want for an independent state cripples the peace process.

U.S. Secretary of State Hillary Clinton announced after Washington's policy change on construction that the United States would push for a return to indirect talks now that direct contacts had failed to produce results.

Senior EU figures, including half a dozen former leaders, sent a letter last week urging the bloc to take a firmer approach with Israel over the Middle East peace process.

The EU has struggled to carve out an influential role in the Middle East peace process and has tried mainly to help the Palestinians strengthen their economy, particularly in the blockaded Gaza Strip.

The EU is the Palestinians' largest international donor, giving hundreds of millions of euros in economic assistance.

On Monday, EU ministers agreed to help Gaza authorities expand the capacity of their border crossings by buying and installing security equipment, training border personnel and building up the road network.

Israel announced a further relaxation of its blockade this month, saying it would permit exports of manufactured goods. It eased import restrictions in June, but the EU says the movement of goods is still inadequate. The EU reiterates its call for the immediate, sustained and unconditional opening of crossings for the flow of humanitarian aid, commercial goods and persons to and from Gaza, the EU foreign ministers said.

Despite some progress following the decision of the Israeli government ... to ease the closure, changes on the ground have been limited and insufficient so far, they said.