But actors' union Equity, which represents the 80 dancers, has ruled out strike action over the grievances.
Talks took place with executive director Tony Hall on Tuesday after reports that dancers were "infuriated" with Mr Stretton, an Australian who took up his post in 2001.
"We have talked about a number of casting and management issues with Tony Hall," an Equity spokesman told BBC News Online.
"He undertook to take away our concerns and come back to us.
"There will be a number of further meetings in coming weeks," said the spokesman. "There's no threat of a strike - that's fantasy."
A spokesman for the Royal Ballet confirmed that more meetings would take place, but declined to say more.
'Unimaginative'
Recent newspaper reports suggested that dancers were fed up with the management style of Mr Stretton, who is said often to keep them in the dark about who will be taking the stage in which productions.
Mr Stretton, 50, is a former director of the Australian Ballet and assistant director of the American Ballet Theatre.
His reign at the Royal Ballet has been criticised for an unimaginative repertoire in which he has failed to stage as many new productions as expected.
The Royal Ballet's cast of principal dancers includes Darcey Bussell, Romanian starlet Alina Cojocaru and French lead Zenaida Yanowsky.
All are Equity members.