MEXICO CITY, June 26 (Reuters) - Crude output from Mexico’s struggling Cantarell oil field fell in May for the eighth month in a row to 1.038 million barrels per day, its lowest level in more than 12 years, Energy Ministry data showed on Thursday.

The fading jewel of Mexico’s oil industry, Cantarell has declined rapidly since 2004 and is pulling down overall oil production in the world’s No. 6 oil-producing nation, threatening Mexico’s status as a top supplier to the United States.

The field’s waning yields are putting pressure on the government to overcome opposition in Congress to its proposal to overhaul the oil industry by lowering barriers to private investment as a way to speed up new drilling projects.

Cantarell for years produced 60 percent of Mexico’s crude oil, and production peaked at around 2 million barrels per day in 2004.

But its May output fell again, from 1.074 million bpd in April, to stand at its lowest level since January 1996, according to Energy Ministry data.

Cantarell, which state oil monopoly Pemex sees declining at around 15 percent annually, accounted for just 37 percent of Mexico’s May oil production, down from 39 percent in April.

Pemex is battling to increase output at less productive fields, such as the offshore Ku Maloob Zaap complex and the technically tricky onshore Chicontepec field.

Pemex also wants to reach potentially huge deep-sea oil deposits in the Gulf of Mexico, but says that without an easing of constitutional laws against taking on foreign partners, it could be 20 years before it is producing deepwater oil.

Energy Minister Georgina Kessel says that without reform, Mexico’s oil output could shrink to half its size by 2021. (Reporting by Catherine Bremer; Editing by Walter Bagley)