By David Milliken

LONDON (Reuters) - Britain's central bank looks on course to hold interest rates at a 16-year high of 5.25% on Thursday as underlying inflation pressures prove persistent, depriving Prime Minister Rishi Sunak of a much-needed boost ahead of a July 4 election.

Bank of England Governor Andrew Bailey opened the door early last month to a rate cut, saying he was "optimistic that things are moving in the right direction" and that a June rate cut was an option - although no fait accompli.

But despite data on Wednesday showing headline inflation fell back to the BoE's 2% target for the first time in nearly three years in May - reaching its goal quicker than in the United States or euro zone - the medium-term picture is now less reassuring.

Services price inflation has fallen less than the BoE expected at the time of the last meeting - only declining to 5.7% rather than 5.3% - and private-sector wage growth is almost twice the rate the BoE judges as compatible with 2% inflation.

Last month the central bank forecast inflation would rise to around 2.6% by the end of the year, as the effect of recent cuts to regulated household energy bills faded.

None of the 65 economists in a Reuters poll last week said they expected the BoE to follow the lead of the European Central Bank and cut rates this month, with the next statement on Aug. 1 looking by far the most probable start date for an easing cycle.

Instead, the expectation is for a repeat of May's 7-2 vote split, when Deputy Governor Dave Ramsden and external Monetary Policy Committee member Swati Dhingra voted for a quarter-point cut.

"We think the Bank of England is left waiting for more reassuring data ... either in the shape of a more decisive moderation in services CPI or with all other broader signals ... pointing in a softer direction," Victoria Clarke, chief UK economist at Santander (BME:SAN), said.

While unemployment is at a two-and-a-half year high of 4.4%, economic growth this year has been reasonable by Britain's recent weak standards.

Financial markets are doubtful about an August rate cut. On Wednesday they priced in only a 30% chance, with a first move more likely in September and a risk of a delay until November, similar to expectations for the U.S. Federal Reserve.

Either way, any cut is likely to be too late for Sunak, whose Conservative Party is around 20 points behind the opposition Labour Party in the pre-election polls.

While Sunak has sought credit for the fall in inflation since he took office in October 2022, when it was at a 41-year high of 11.1%, Labour blames high mortgage rates on economic mismanagement by the Conservatives' previous leader, Liz Truss.

Since the start of the election campaign the BoE has been in a self-imposed period of silence, cancelling public events.

Before that, BoE Chief Economist Huw Pill had described an excessive focus on a June rate cut as "ill advised" but both he and Deputy Governor Ben Broadbent - who steps down at the end of this month - said a rate cut over the summer was possible.

The BoE began to raise rates in December 2021, earlier than other major central banks, and they reached their current peak in August 2023.