The Nigerian winger, still just 22 years of age, joined the Toffees ahead of the 2017-18 season for a fee around £7.5m after a 2016-17 season that saw him score 22 goals for Eupen in the Belgian top-flight.
However, Onyekuru never played in a competitive fixture for the Premier League side, as he was immediately loaned out to Anderlecht and scored 10 goals in all competitions for the Belgian side before a knee injury cut his campaign short.
He was loaned out again for the 2018-19 season, scoring 14 league goals to help Galatasaray secure a Turkish domestic double.
Writing on Twitter, Onyekuru thanked Everton for his opportunity as well as hailed his experience with Galatasaray last season.
Onyekuru wrote: "I want to thank Everton for the opportunity and making it possible for me to experience the best year of my career at Galatasaray.
"My time in Turkey was nothing short of amazing. The supporters, the club and my team-mates made me feel at home from day one."
AS Monaco are delighted to announce the signing of Henry Onyekuru from Everton. Our new attacking midfielder joins us on a 5-year deal . pic.twitter.com/5r6drWUoDQ— AS Monaco EN (@AS_Monaco_EN) August 12, 2019
Last season saw head coach Leonardo Jardim sacked for Thierry Henry after a dreadful start to the campaign left them in the relegation zone.
However, Henry failed to reverse their fortunes and after crashing out of the Champions League and remaining under threat of the drop, the former Arsenal great was replaced, with Jardim returning to the role.
Jardim did just enough to keep Monaco up, with the club surviving thanks to a 17th place finish by just two points over Dijon.
Their start to the 2019-20 campaign did not offer much evidence a corner had been turned, as the club suffered a heavy 3-0 defeat to Lyon in the opening round.
Onyekuru could make his debut for the club when they travel to Metz on Saturday, as they look to secure only their second win in their last 10 Ligue 1 contests.