The 19-year-old centre-back signed his first professional contract with the Red Devils in 2018, but started chemotherapy in February of this year.
After successfully undergoing treatment, Taylor joined the wider group for full training for the first time this week and will continue to be involved intermittently as he steps up his return to the full-time programme.
He told the official club website: "I feel immensely proud and happy to be back on the training pitch with my team-mates and friends.
"Without the invaluable support of my family, team-mates and, of course, the nurses and doctors, my return to training at this stage would not have been possible.
"I want to thank all of the club's medical and sports-science staff that I have worked with throughout my rehabilitation.
"The next step is to continue working hard in the gym, and out on the grass, to hopefully make my return on the pitch."
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Max Taylor back in United training after cancer treatment
Unai Emery insists Mesut Ozil needs rest despite playing just 71 minutes of football this season as the Arsenal head coach left the forward at home for the Europa League tie with Eintracht Frankfurt.
Ozil, 30, had not featured at all this season until he started Sunday's 2-2 Premier League draw at Watford.
He missed the start of the season after being the target of an attempted carjacking alongside team-mate Sead Kolasinac, while illness also saw him miss out.
Ozil regained his fitness but was still left on the sidelines until the weekend.
He retired from international football with Germany last year amid controversy and, on what would have been his first return to play in the country, he would have no doubt been in for a hostile reception from a fervent Frankfurt crowd.
But instead Emery offered up the rather bizarre excuse that Ozil was instead being rested alongside Sokratis Papastathopoulos – who has played every minute of the opening five Premier League games of the season.
"We are playing now a lot of matches and I want to use different players and also can have some rotation in the first 11 and in the 19 players in the squad for tomorrow and Sokratis and Mesut they rest this match," he said.
"I prefer after the Sunday match to rest that match and to train and to be ready for Sunday.
"Because we are going to play another match on Sunday. My decision is because he played and also I'm thinking tomorrow the focus and the first 11.
"After that I decided who are the players who can be the best for tomorrow and also the players not and that is Sokratis and Mesut.
"Really the first match he played on Sunday, he played well and was good physically – also giving us his capacity and skills and after his first minutes (of the season) it was good for us.
"Now he is ready to help us, not tomorrow but on Sunday – we are going to play another match and tomorrow we are thinking to do some rotation."
There will now be plenty of attention on whether Ozil starts Sunday's league visit of Aston Villa, given he has been left out here.
He was front-page news after carjackers targeted him and Kolasinac in Golders Green on the eve of the new Premier League campaign.
Kolasinac spoke about the incident for the first time and admitted it was tough to deal with but is now keen to move on.
"About what happened I don't want to talk about it in details," he said.
"It's happened and it's something that belongs to the past. The same with Mesut. We are back and we focus on our job and team. We have a very important match ahead here in Frankfurt.
"Well of course I'm happy to be back. Of course mentally this was hard, missing also training for a couple of days.
"I made up for it. I'm back and happy to be back with the team and able to perform with them on the pitch with the team."
Spurs' run to last season's final was littered with great escapes and last-ditch turnarounds but their group stage this term looked like getting off to a winning start in Athens.
They were not particularly worthy of it, but a Harry Kane penalty and Lucas Moura's thunderbolt put them 2-0 ahead after 30 minutes.
But Daniel Podence's goal on the stroke of half-time let Olympiacos back into it and Mathieu Valbuena's spot-kick after the restart gave the hosts a deserved point.
It confirmed boss Mauricio Pochettino's assertion that Spurs would not be among the favourites to win this season's competition and though the draw means they will still be expected to get out of this group – which also features Bayern Munich and Red Star Belgrade – with relative ease.
They could have laid down a big marker towards that goal, but instead will be heading back to London on the back of a difficult night and with a 12.30pm kick-off on Saturday at Leicester to prepare for.