BUDAPEST, July 11 (Reuters) - Hungary's central bank will not accelerate cuts in its one-day deposit rate despite the likelihood of a fall in inflation into single digits by the autumn, Deputy Governor Barnabas Virag said on Tuesday.

The National Bank of Hungary cut its one-day deposit rate by another 100 basis points to 16% last month to ease the burden on the stagnating economy, as the European Union's highest inflation rate is finally slowing and the forint has gained.

The bank launched the emergency one-day deposit rate, the highest in the EU, in October to shore up the falling forint amid a surge in inflation, but started paring back emergency rate rises in May.

Headline inflation fell to 20.1% year-on-year in June and the trend is expected to accelerate over the coming months, Virag said. He added, however, that the bank must stick to a gradual and cautious path of policy normalisation.

"It can be ruled out that over the coming period monetary policy will continue this process of policy normalisation at a faster pace," Virag told private broadcaster InfoRadio in an interview.

"There is no reason to justify any deviation from the policy path that the Monetary Council embarked on in May."

He added that the central bank should pursue a monetary policy that ensures disinflation continues next year, when economic growth is expected to accelerate, while real wage growth could return into a positive range already in late 2023.

The continued stability of the forint, which has gained some 5% versus the euro this year, will be another key factor in supporting the fall in the inflation rate, Virag said.

"The foreign exchange market is a lot more stable, and this is needed and will be needed going forward to ensure that we keep inflation on a sustained downward path towards our price stability objectives," Virag said.

"We will conduct our monetary policy accordingly." (Reporting by Gergely Szakacs Editing by Mark Potter and Mark Heinrich)