The announcement was issued by Kirsten McMillan, a studio animal trainer who has worked with animals on many projects for American TV networks including ABC and CBS.
Her role also consists of looking after Roscoe while Hamilton is travelling the globe on F1 duty, and her social media accounts contain various snaps of the French Bulldog for fans to enjoy.
Hamilton boosted by good news
In April, McMillan provided details of treatment Roscoe was receiving after contracting pneumonia, revealing that he was scheduled to make a trip to the vet.
And she has taken to Instagram once again to keep everyone abreast of the latest information on Roscoe's recovery.
Writing on her story, she said: "Hi guys! For those asking: Roscoe is nearly fully recovered.
"Haven’t been posting our silly adventures because he's been with his dad, so his dad will be the one with the updates guys.
"Just know he's been looking well and recovering really nicely thankfully."
Hamilton adopted Roscoe back in 2013, and the pair have been spotted together in the paddock on race weekends over the years.
Last season, Roscoe was seen wearing Ferrari colours on his own social media pages ahead of Hamilton's move to the Italian giants.
However, things have not gone according to plan for the former Mercedes man, who has cut a frustrated figure throughout much of his time in red.
He heads into this weekend's Miami Grand Prix seventh in the drivers' standings, behind team-mate Charles Leclerc and Kimi Antonelli, who replaced him at the Silver Arrows.