Adebayor has beenwithout a club since December after terminating his contract with Kayserispor, ending his three-month spell at the Turkish side.
"Hello to the Olimpia fans".
Adebayor will be reunited with former Manchester City team-mate, Roque Santa Cruz, who joined the club in 2016.
"He was attracted by the challenge of playing in Copa Libertadores, where he has never competed", Olimpia club president Marco Trovato said.
Adebayor celebrates in front of the Arsenal fans after scoring for Manchester City
Adebayor could be followed to South America by Ivorian midfielder Yaya Toure.
After joining league rivals Manchester City, Adebayor spent a year and a half with the Cityzens before heading to Real Madrid to win the 2011 Spanish Cup.
Olimpia is Adebayor's 10th professional club since leaving his native Togo in 2001 to head to France, where he featured for Metz and Monaco before sealing his big move to Arsenal.
Since then, the striker has represented Crystal Palace, Istanbul BB and Kayserispor.
In 446 league appearances spanning four different countries, Adebayor has 161 goals to his name and 52 assists.