WASHINGTON, June 9 (Reuters) - U.S. existing home sales increased more than expected in May, though rising mortgage rates and still-tight inventory remain a challenge for the housing market.

Home sales jumped 3.2% last month to a seasonally adjusted annual rate of 4.170 million units, the National Association of Realtors said on Tuesday. Economists polled by Reuters had forecast home resales advancing to a rate of 4.07 million units.

Sales increased in the Northeast, South and Midwest regions, but were unchanged in the West. Home resales, which are counted at the closing of a contract, increased 3.2% year-on-year in May.

"More Americans are on the move, with home sales rising to the highest level since December," said Lawrence Yun, the NAR's chief economist. "This is great news for the housing market."

Last month's sales likely reflected contracts signed in March and April. Mortgage rates started rising in March as the U.S. and Israel attacked Iran, before easing towards the end of April following a ceasefire. The Middle East conflict is fanning inflation, through higher prices for energy and other products that are shipped through the Strait of Hormuz. That has helped to lift U.S. Treasury yields, which mortgages track.

The average rate on the popular 30-year fixed-rate mortgage has increased by about 50 basis points since the war started at the end of February. With prospects of an interest rate cut from the Federal Reserve fading amid rising inflation and a resilient labor market, mortgage rates are likely to remain elevated.

The government is expected to report on Wednesday that the Consumer Price Index surged 4.2% on a year-over-year basis in May, a Reuters survey of economists predicted, which would be the largest gain since April 2023. The CPI rose 3.8% in April.

The NAR's housing affordability index improved to 105.6 in May from 97.5 a year ago. Inflation is outpacing wage growth. The median existing home price last month rose to $429,300, up 1.3% from a year ago.

The inventory of existing homes increased 3.3% to 1.55

million units. Supply, which typically increases in May, remains well below pre-pandemic levels. Supply was up 0.6% from a year ago. At May's sales pace, it would take 4.5 months to exhaust the current inventory of existing homes, down from 4.6 months a year ago.

The median number of days on the market for listed properties increased to 29 from 27 a year ago. First-time buyers accounted for 35% of sales, up from 30% a year ago. Economists and realtors say a 40% share in this category is needed for a robust housing market.