BRUSSELS, June 21 (Reuters) - European Union leaders will seek "swift and decisive" progress by the end of the year on a U.S.-style capital market to steer private investment into the bloc's green and digital economy, according to draft conclusions of next week's EU summit.

The leaders, who will meet in Brussels on June 27-28, agreed in April to reforms on several fronts to revitalise the EU economy and help it catch the United States and China in the global tech race.

Key to that could be a EU capital markets union (CMU), but negotiations on it have stalled for nearly a decade because EU members do not want to relinquish control of national financial rules. EU finance ministers have committed to setting up a CMU by 2029.

According to the draft conclusions, which could be changed before or during the summit, the leaders will call for accelerated work towards that goal.

"The European Council looks forward to swift and decisive progress by the end of the year," the draft says.

The aim of the CMU is to harmonise laws such as on capital gains tax and bankruptcy to create a single capital market, instead of 27, and to persuade households to invest in securities rather than park their savings in bank accounts.

The European Commission, the EU's executive arm, says Europe will need 650 billion euros ($695 billion) - around 4.5% of its economy - of extra investment a year until 2030 to compete in the global green and digital transition.

That, it argues, can only come from the private sector. ($1 = 0.9355 euros) (Reporting by Philip Blenkinsop and Julia Payne Editing by Christina Fincher)