* EU stocks and U.S. futures trade roughly flat

* Oil prices fall as Iran and U.S. say progress made in

talks

* Sterling and UK bonds rise after PM Starmer resigns

(Updates prices)

By Wayne Cole and Harry Robertson

SYDNEY/LONDON, June 22 (Reuters) - European stocks and U.S.

futures steadied while oil prices dipped on Monday after

mediators said progress had been made in U.S.-Iran peace talks,

helping calm fears the fragile process to end the war was

breaking down.

Meanwhile, the pound and UK bonds rose after Prime Minister

Keir Starmer announced his resignation, paving the way for

Britain's seventh leader in 10 years.

The Iran war talks had earlier been overshadowed by Tehran's

announcement it had again closed the Strait of Hormuz, prompting

U.S. President Donald Trump to threaten fresh attacks.

But officials from Qatar and Pakistan said progress was made

on a roadmap to reach a final deal in 60 days. That was backed

up by U.S. Vice President JD Vance, who said Tehran had agreed

to allow nuclear inspections.

The apparent progress in discussions saw Brent crude

futures shed early gains to ease 1.8% to $79.07 a barrel, far

below its May peak of $126.41.

Europe's STOXX 600 index wavered and was last up

0.15%, while U.S. S&P 500 futures pared early losses to

trade 0.1% lower.

"There does appear to be further progress being made during

talks in Switzerland towards a lasting settlement, and oil

prices have dipped again," Susannah Streeter, chief investment

strategist at Wealth Club, said of the Iran talks.

"It is clear there is still a long way to go, and more

obstacles may emerge before a long-term deal is signed."

Asian stocks climbed overnight, supported by the apparent

progress in peace talks. Japan's Nikkei rose 1.6%, while

South Korea's red-hot market added 0.7%, after surging

more than 11% last week on demand for semiconductor stocks.

STARMER RESIGNATION CLOUDS UK OUTLOOK

The pound reversed earlier losses to trade flat at $1.324

while gilts rose on Monday after Starmer announced his

resignation, which had been widely rumoured over the weekend.

Former Manchester Mayor Andy Burnham is the favourite to

succeed Starmer, but investors said a key question for nervy UK

bond markets would be who becomes finance minister.

"A new leader does not fundamentally alter the difficult

fiscal situation they're going to inherit," said Nick Rees, head

of macro research at Monex Europe.

The euro eased 0.15% to $1.146, after hitting a

three-month low on Friday at $1.1418.

Treasuries remained under pressure following a hawkish turn

by the Federal Reserve last week that led markets to price in a

75% chance of a rate hike as early as September.

Futures imply around 38 basis points of tightening by

year-end, while yields on 2-year notes rose as much

as 4 basis points to the highest since early 2025 at 4.230%.

"Our baseline call is for patience and a first hike in the

second half of 2027, but (we) believe the margin for error and

the tolerance for further inflation is limited, with genuine

risks of earlier hikes," said Fabio Bassi, head of cross-asset

strategy at JPMorgan.

The Fed's hawkish outlook helped push the dollar up 0.3% to

161.71 yen, with only the threat of Japanese

intervention preventing the currency rising to 2024's 40-year

high of 161.96.