Argentine forward Becchio was the club’s top scorer last season with 20 goals, but missed the end of the campaign after a hamstring injury.
The 27-year-old returned in time for light pre-season training, but has since ruptured a tendon and now faces an operation.
Meanwhile, South African Somma faces being out of play for six to nine months with a serious knee injury.
The Bafana Bafana international, 26, sustained the cruciate injury after a knock in pre-season training.
The pair join defender Andy O'Brien - suffering from a side strain - on the injured list as Leeds step up their pre-season friendlies.
"It's been frustrating for all of them," United boss Simon Grayson told the club’s website. "Andy has worked hard over pre-season, Dave's was innocuous, and Luciano had the injury at the back end of last season and has done some work with us."