* U.S. job growth surges, unemployment rate near six-year low
* Asian stocks, dollar rise on upbeat U.S. jobs data
* Iraqi Kurds say will sue Baghdad if it blocks oil sales
* Islamic State seizes oil field and towns in Syria's east
SINGAPORE, July 4 (Reuters) - Brent futures held steady around $111 a barrel on Friday on signs of an improving demand outlook, although the benchmark is still set to post its biggest weekly loss since early January as supply worries that have rattled oil markets for weeks recede.
U.S. oil futures are down for a seventh straight day amid expectations of more shipments from Libya and Iran. The last time the contract fell for so many days was in December 2009.
Yet hopes of an improved demand outlook in the United States and China and worries that the Iraqi crisis may spin out of control mean losses will be limited.
Brent crude rose 7 cents to $111.07 a barrel by 0453 GMT and looks set to lose 2 percent this week.
U.S. oil lost 7 cents to $103.99 a barrel and is set to end the week down 1.7 percent, the biggest weekly loss since end-April.
"Supply fears are easing somewhat, but Iraq is setting a high floor on prices," said Victor Shum, vice-president of energy consultancy IHS Energy Insight. "It's a combination of improving demand outlook and supply worries. One would have to be really brave to sell in this market."
The twin factors will keep the U.S. benchmark above $100 a barrel, Shum said.
Investors are watching the progress of talks between Tehran and world powers to end the decades-old dispute over its controversial nuclear programme. Iran has reduced demands for the size of its future nuclear enrichment programme although the West is urging Tehran to compromise further.
Prospects of further easing of sanctions and more Iranian oil hitting the market come a day after Libyan Prime Minister declared an end to an oil crisis that has cut exports from the OPEC member to a trickle.
"If it wasn't for the situation in Iraq, with the possibility of higher Libyan supplies, more Iranian oil, we would have seen prices a lot lower," Shum said.
On the demand front, U.S. employment growth jumped in June and the jobless rate closed in on a six-year low, pointing to decisive evidence the country was growing briskly heading into the second half of the year.
The positive signs of growth from the United States closely followed data from China that showed factory activity there hit multi-month highs in June, adding to sentiment that demand for oil would remain solid.
Broader financial markets - including Asian shares and the dollar - firmed after the U.S. jobs data report.
But investors are still nervous about the unfolding crisis in OPEC's second-largest producer Iraq. Its autonomous Kurdish region has hit back at Baghdad over independent oil exports, with the Kurdistan Regional Government (KRG) threatening in a letter to countersue the central government for trying to block its sales.
Militants from the Islamic State group also seized Syria's largest oil field from rival Islamist fighters on Thursday, strengthening its advance across the eastern Deir al-Zor province, an opposition monitoring group said.
However, the change in control of the field should have little impact on global oil markets. Syria is not a significant producer and has not exported any oil since late 2011, when international sanctions took effect to raise pressure on President Bashar al-Assad.
(Reporting by Manash Goswami; Editing by Tom Hogue)