By Julia Fioretti

BRUSSELS (Reuters) - The European Union's digital chief has said that failure to solve the last remaining barrier to abolishing mobile roaming charges across the bloc in June would lead people to question its ability to deliver on promises.

EU lawmakers and member states hold a third and probably final round of talks on Tuesday on where to set caps for the wholesale roaming charges telecom operators pay each other when their customers call, send texts or surf the web abroad.

It is the last piece of the puzzle needed to enable the abolition of retail roaming charges in June, crowning a decade of efforts by Brussels to allow citizens to use their phones abroad without paying extra.

The effort took on an added significance after Britain voted to quit the bloc last year in a surge of anti-EU sentiment and Brussels has sought to show it works for ordinary citizens.

But the two sides remain far apart on where the wholesale caps for data should be set, with the European Parliament pushing for an initial cap of 4 euros ($4) per gigabyte while member states want it to start at 8.5 euros per gigabyte.

European Commission Vice President Andrus Ansip wrote to the two sides on Friday urging them to show "significant flexibility" to achieve a final agreement.

"If no political compromise can be achieved next Tuesday, people will rightly question our common will and ability to deliver on our promise to them. That is a risk we should not run," Ansip wrote.

The split on wholesale roaming caps stems from wide differences in domestic prices and travel patterns across the bloc.

Countries in northern and eastern Europe with low domestic prices and generous packages favor lower wholesale caps to avoid companies raising prices in their home markets, effectively making poorer customers subsidize frequent travelers.

However countries in the tourist-magnet south worry that their operators could be forced to hike domestic prices to accommodate the seasonal tourist traffic. They also fear operators will put off investment in networks if foreign operators can gain cheap access to their infrastructure and undercut them domestically.

(Reporting by Julia Fioretti Editing by Ruth Pitchford)