The Prestige - Greek-owned and registered in the Bahamas - is thought to have been carrying 77,000 tons of oil.
It is reported to have a crack in one of its oil tanks.
Rescue workers succeeded in getting a line aboard the ship after 16 failed attempts, in gale force winds.
The European Union has said it is concerned Britain did not carry out proper checks on the tanker.
New rules
The office of EU Transport Commissioner, Loyola de Palacio, has sent British authorities a letter questioning whether it carried out the required inspections of the tanker when it docked in Gibraltar before its latest journey from Latvia.
Mr Palacio said, under new rules, port authorities had to check 25% of all ships coming into dock and added that such inspections should be especially targeted at ships with a history of problems.
The Prestige, which was built in Japan in 1976, last underwent inspection in 1999.
Rescue workers evacuated most of the crew from the Prestige after it got into trouble in rough seas and high winds on Wednesday, flying them to the port of Vigo.
Crew rescued
The first seven crew members rescued, all from the Philippines, said that they felt a bump and the ship then began taking on water.
Spanish maritime officials, have expressed fears that the tanker may sink.
Environmentalists have warned that the area, one of Spain's most scenic regions, could face ecological catastrophe if the tanker continues to spill its cargo.