He resumed baseball activities on Saturday, taking batting practice for the first time since sustaining a concussion on a diving catch a week ago.
Sweeney is eligible to be activated from the seven-day disabled list Monday.
Darnell McDonald (right oblique strain) is also close to a return. McDonald took a day off on Saturday after starting his Minor League rehab assignment for Triple-A Pawtucket. He will play again for the PawSox on Sunday and Monday.
If all goes well, McDonald could be activated by the Red Sox on Tuesday.