* Pound steady as investors monitor US-Iran peace talks

* Britain remains vulnerable to oil price swings

* BoE Governor Bailey signals no rush for rate hikes

By Samuel Indyk

LONDON, June 1 (Reuters) - The British pound was little

changed against the dollar on Monday, remaining within its

recent narrow range as investors awaited progress on peace talks

between the U.S. and Iran.

Negotiators from the U.S. and Iran are still working on a

deal, but fresh attacks in the Gulf have challenged optimism

about the reopening of the Strait of Hormuz, the key oil and gas

artery that has been closed to most maritime traffic since the

start of the war in February.

Oil futures jumped on Monday but fell almost 20%

last month, the biggest monthly drop since March 2020.

Britain's reliance on energy imports leaves it more exposed

than the U.S. to higher fuel costs. While oil prices have fallen

in recent weeks, they remain 30% higher than before the war.

On Monday, the pound was broadly flat against the

dollar at $1.3460. Against the euro it was up 0.1% at

86.56 pence.

"The pound has been quite resilient during the war,"

said Tommy von Brömsen, FX strategist at Handelsbanken.

"What has supported the pound is the level of interest

rates. Britain has a higher interest rate than most other

countries."

Prior to the war, the Bank of England had been expected to

lower interest rates twice this year but markets quickly moved

to price in rate hikes as investors believed that the BoE would

need to take action to combat energy-driven inflation.

Markets are now pricing in about 35 basis points of monetary

policy tightening this year, implying one quarter-point hike and

around a 40% chance of a second.

On Friday, BoE Governor Andrew Bailey said that allowing

inflation to run above the central bank's 2% target was

justified, signalling there was no need to move quickly with

interest rate hikes.