PARIS/FRANKFURT, April 16 (Reuters) - European Central Bank policymakers continued to make the case for an interest rate cut in June on Tuesday as inflation remains on course to ease back to 2% by next year, even if the path for prices still proves bumpy.

The ECB held rates at a record high last week but opened the door wide open to a June rate cut, batting back doubts about its resolve after market expectations retreated on unexpectedly high U.S. inflation figures.

"Barring major shocks and surprises, we should decide a first rate cut in early June, followed by others in a pragmatic and agile gradualism," French central bank chief Francois Villeroy de Galhau said in New York.

ECB President Christine Lagarde echoed that message, arguing that inflation is easing much as the ECB had expected and fresh turbulence in the Middle East has so far had little impact on commodity prices.

"Subject to no development of additional shock, it will be time to moderate the restrictive monetary policy in reasonably short order," Lagarde told CNBC. "We are observing a disinflationary process that is moving according to our expectations."

Markets now see 77 basis points of rate cuts this year, or three moves - a big change compared to six weeks ago when over a 100 basis points were priced in.

Although the ECB has not officially said it will cut in June, policymakers have repeated the reference to the month so often that the ECB has essentially pre-committed, and walking back would damage credibility, economists say.

"This assumes there will be no further setbacks, for instance in the geopolitical situation and therefore in energy prices," Finnish central bank chief Olli Rehn said in Helsinki.

Although oil prices have increased and the euro has weakened in recent weeks, the changes have been too modest to make a significant impact on consumer prices.

Domestic inflation, particularly from services and wages, will have a bigger impact as will base effect, when big energy price falls from a year ago get knocked out of data later this year.

"Inflation rates might go down, might go up a little bit, but they will fluctuate around this line that, for the moment, is going downwards," Lagarde said. (Reporting by Leigh Thomas and Balazs Koranyi; additional reporting by Anne Kauranen and Rishabh Jaiswal; Editing by Sharon Singleton and Mark Heinrich)