Bay Area media outlets reported Strickland will begin serving his suspension with Monday night's game at Atlanta. He would be eligible to return June 26 against the Chicago Cubs.
Strickland's punishment was levied almost three weeks ago but was delayed pending Strickland's appeal. He also was fined.
Strickland and Nationals star Bryce Harper were suspended after the Giants reliever hit Harper in the hip with a pitch in the eighth inning during the Nationals' 3-0 win May 29 at San Francisco. The Washington slugger then charged the mound, wildly firing his helmet before trading punches to the head with Strickland.
MLB chief baseball officer Joe Torre's explanation of the disciplinary decisions said Strickland intentionally hit "Harper with a pitch, inciting the bench-clearing incident and fighting," while Harper's suspension came "for charging the mound, throwing his helmet and fighting."
The bad blood stems from a pair of home runs Harper hit off Strickland in the 2014 playoffs, when the Giants went on to capture their third World Series championship in five years.
After the first homer, Strickland said he wasn't afraid of Harper. After the second, Harper stared at Strickland as he rounded the bases.
The Associated Press contributed to this report.