By Sara Sjolin, MarketWatch

LONDON (MarketWatch) -- European stocks pushed higher on Friday, driven by optimism over U.S. debt talks, with the benchmark index for the region posting its first weekly gain in three weeks.

The Stoxx Europe 600 index climbed 0.4% to close at 311.61, ending the week 0.6% higher.

On Thursday, the index rallied 1.7% on signs U.S. lawmakers were moving closer to agreeing on a deal to lift the debt ceiling. The U.S. government, meanwhile, has been partially shut down for 11 days after Congress failed to agree on a budget for the fiscal year that began Oct. 1.

"The markets have been desperate for some good news for a while. The optimism from yesterday carried on overnight and the markets remained on positive footing today," said Richard Perry, chief market strategist at Central Markets.

"But the longer we get to the debt-ceiling deadline without a deal, the more risk could creep back into the market and it could be a bit of a funny week. When they get it resolved, we could have a big rally. With [Janet] Yellen moving into the Fed, it suggests that tapering will not happen this side of 2014 and once the debt ceiling is taking out, there should be more jet fuel to push markets higher," he added.

House Republicans on Friday offered the White House a debt-ceiling increase and an end to the government shutdown in exchange for spending cuts, the Associated Press reported.

The offer came after House Republicans on Thursday proposed a six-week extension to the borrowing limit, in efforts to avoid a default on Oct. 17 -- the date Treasury Secretary Jack Lew said is the deadline for raising the debt limit. House Republican leaders and President Barack Obama ended a roughly 90-minute meeting at the White House late Thursday without a deal to raise the debt limit or reopen government, but with a plan to keep talking.

U.S. stocks rallied on Thursday, with the Dow Jones Industrial Average (DJI) putting in its best one-day point gain since December 2011. Wall Street stocks also traded with gains on Friday.

"[There's] nothing concrete, but the fact they are talking is a step in the right direction compared [with] the recent flat refusals to give any ground. [Are] Republicans finally softening, after being apportioned with more of the blame?" said Mike van Dulken, head of research at Accendo Markets, in a note to clients.

Among country-specific indexes in Europe, the U.K.'s FTSE 100 index rose 0.9% to 6,487.19, and ended the week up 0.5%.

Germany's DAX 30 index added 0.5% to 8,724.83, closing at the highest level on record and mastering a 1.2% advance on the week.

France's CAC 40 index closed slightly higher at 4,219.98 and rose 1.3% on the week.

Shares of Royal Mail PLC soared 38% to 4.55 pounds ($7.26) on its first trading day in London. The initial-public-offering price was GBP3.30 a share, and the solid interest on the first trading day fueled suggestions that the previously state-owned company was sold off too cheap.

Elsewhere, shares of Royal Philips NV gained 2.9% after Goldman Sachs lifted the electronics firm to buy from neutral. The analysts said they saw the Dutch firm as one of a few remaining stocks with re-rating potential within "capital goods." "We expect its steady progress toward higher margins and returns, better cash quality and more predictable earnings to be rewarded through multiple expansion," they said.

On a more downbeat note, shares of Geberit AG lost 1.6% after Goldman Sachs cut the toilets and sanitary piping-system maker to sell from neutral on concerns it will "struggle to outgrow the underlying market in its most profitable segment."

Shares of Neste Oil Oyj slid 7.5%. UBS lifted shares to neutral from sell, but also said it sees net income flat over 2014-17.

Subscribe to WSJ: http://online.wsj.com?mod=djnwires