The 21-year-old defender has been on the books of the Red Devils since he joined their academy at the age of seven, but he has only made one senior appearance for the club - a brief substitute outing in a Premier League game against Wolverhampton Wanderers in May 2021.
Fish has spent the last two seasons on loan in Scotland with Hibernian, making 66 first-team appearances in all competitions, but he is now seeking a permanent exit from Old Trafford to further his career.
According to the Manchester Evening News, Man United have agreed to let Fish join Cardiff on a permanent basis, with the deal set to be completed at some stage this week.
Fish is said to have attracted interest from several second-tier clubs who were eyeing up both loan and permanent transfers, but it is Cardiff who has won the race for his signature.
Cardiff's move for Fish comes at a time when fellow defender Mark McGuinness is reportedly closing in on a £10m transfer to Luton Town.
Leeds to step up Jack Clarke pursuit?
Staying in the Championship and Leeds United are reportedly planning to sign two attackers before the summer transfer window closes, with Sunderland's Jack Clarke emerging as one of their targets.
The West Yorkshire outfit have funds to spend after receiving £40m from the sale of Georginio Rutter to Brighton & Hove Albion on Monday, while around £75m has been generated from the sales of Crysensio Summerville, Archie Gray and Glen Kamara earlier this summer.
In a bid to replace Rutter and Summerville, two key attackers from last season, Leeds will prioritise reinforcements in the final third before the transfer window shuts on August 30, according to Football Insider.
One target said to be on their shortlist is highly-rated Sunderland winger Clarke, who spent nine years in Leeds' academy and made his senior debut for the Whites before moving to Tottenham Hotspur in 2019.
Clarke has since excelled as a star player for Sunderland, scoring 15 goals in 40 Championship games last season, and has subsequently been linked with a number of Premier League clubs as well as Leeds.
The 23-year-old, who has already scored one goal and provided one assist in Sunderland's opening two second-tier matches of this season, has two years remaining on his contract at the Stadium of Light.
Ipswich sign Cajuste after Brentford collapse
Elsewhere, Ipswich Town have completed the signing of midfielder Jens Cajuste from Napoli on loan until the end of the season.
The 25-year-old was strongly linked with Brentford earlier this month, but a move to the Bees ultimately collapsed and Ipswich quickly swooped in to secure the services of the Sweden international on a temporary basis.
"This feels great and I'm happy to be here," Cajuste told TownTV. "It took a little while with the negotiations but I'm finally here and I'm very excited.
"I've had great conversations with the manager (Kieran McKenna) and this seems like a very interesting project. It's an ambitious club and a team that is progressing.
"I'm a player that works hard and I want to contribute in attack and defence. I want to play as much as possible and I'll be looking to get some good tackles in!"
Cajuste, who played 35 times for Napoli in all competitions last season, becomes Ipswich's ninth signing of the summer and could make his debut for the Tractor Boys when they face Premier League champions Manchester City at the Etihad Stadium on Saturday.