Avelino signed a minor-league contract with the Dodgers on June 9.
Avelino signed a minor-league deal with the Cubs on Friday that includes an invitation to major-league spring training, Jordan Bastian of MLB.com reports.
The Giants released Avelino on Monday, John Shea of the San Francisco Chronicle reports.
Avelino cleared waivers Sunday and will report to the Giants' alternate training site, Alex Pavlovic of NBC Sports Bay Area reports.
Avelino was designated for assignment by the Giants on Sunday.
Avelino was optioned to Triple-A Sacramento on Tuesday, Maria I. Guardado of MLB.com reports.
Avelino was recalled from Triple-A Sacramento on Sunday.
Avelino was optioned to Triple-A Sacramento on Monday.
Avelino (leg) is listed as one of the Giants' available reserves for Wednesday's exhibition game against the Rangers, John Shea of the San Francisco Chronicle reports.
Avelino was removed from Sunday's game against the Rockies due to discomfort in his right leg, Kerry Crowley of The San Jose Mercury News reports. He went 1-for-4 with a double prior to exiting.