Asofa-Solomona attended Storm training on Thursday, two months after his expected return to the club for the start of pre-season was delayed due to a stand-off over his vaccination status.
Melbourne refused to confirm to AAP if Asofa-Solomona was vaccinated against COVID-19 or had received an exemption.
In a statement they said the New Zealand international had "provided all necessary information and certification to comply with the requirements of the Victorian and Queensland governments to allow him to return for training and to play".
Under state rules in Victoria and Queensland, athletes who train and play in public facilities are required to be double vaccinated or must present a valid exemption.
Exemptions are only available if the person in question has a contraindication, has an acute medical condition including a COVID-19 infection where hospitalisation was required or is participating in a vaccine trial.
Unlike the AFL, the NRL has not mandated vaccination.
Asofa-Solomona had previously questioned mandatory vaccination on social media and had spent the past two months training on his own.
"It was challenging doing stuff by myself," the 25-year-old said.
"I spent time playing basketball with these little Filipino guys and trying to make training fun.
"I got a good sweat up and did a bit of cardio and a lot of weight training.
"I had a talk with Bellza (coach Craig Bellamy) yesterday and I'm stoked to be back and I'm 100 per cent on board this year.
"I feel like I'm ready (to play), it's just up to Bellza and we'll see what he says."
Asofa-Solomona's return to training ends speculation that the towering forward would seek a termination of his contract and make a move overseas to rugby union.
Recent changes to borders rules and COVID-19 protocols for athletes in Japan and France made any likelihood of a switch of codes more difficult.
The vagueness surrounding Asofa-Solomona's vaccination status is similar to that of North Queensland captain Jason Taumalolo.
Cowboys coach Todd Payten this week refused to reveal whether the Tonga international was vaccinated.
"We are compliant with NRL protocols and then government health orders," he said.
"I'm excited to be able to choose from a full squad."
NRL chief executive Andrew Abdo said the game was nearing a 100 per cent vaccination rate among players.
"We were criticised for not making this mandatory and we've proven we didn't need to; the players have done the right thing and we're up to like 98, 99 per cent having their vaccinations," he said.Â
Meanwhile, Storm back-rower Kenny Bromwich has joined brother Jesse and Felise Kaufusi in signing with the Dolphins for their inaugural campaign in 2023.
The 30-year-old forward has signed a three-year deal with Wayne Bennett's Redcliffe-based side.
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