The Sunday Mirror reports United plan on sealing a £20m move for Rodon - on the advice of Wales boss Ryan Giggs.
Rodon's fine form saw him make his Wales debut for Giggs' senior side back in September, on the back of breaking into the Swans first team during 2018/19.
One of the regional newspapers covering the club's assault on the Championship table this season labelled 22-year-old Rodon a "special talent" this week - and said he was the natural fit on the conveyor belt of young Welsh talent to join "bigger clubs" in the Premier League.
Who is Joe Rodon?
Born and raised in Swansea and a fan of his boyhood club, Rodon joined the Swans' academy in 2005 as an eight-year-old.
"I sat in these classrooms dreaming of playing for my team and Wales," he said on a trip back to his first school, Llangyfelach Primary School, earlier this year. "I've done both, which is incredible, and this school was the first part of the journey."
Having realised the first step to his dream, he rose through the youth ranks before signing Cheltenham Town on loan in 2018.
There Rodon struggled to make a significant impact at Whaddon Road, starting only seven times in half a season with the League 2 side, although manager Gary Johnson reserved praise for the youngster, saying: "We have been keeping an eye on him for some time and he is undoubtedly a quality player."
On his return to the Liberty Stadium, however, he made his breakthrough in 2018/19, and after a first start in a 1-0 win over Preston in August, went on to start 23 league games during that season.
Barely a month after making his senior debut, then-manager Graham Potter was already publicly urging Swansea to tie Rodon down to a longer contract in south Wales, having been impressed with the start he had made to life as a first-team player.
"I think we need to do it quickly and that's something we're doing behind the scenes," he said in October 2018. "He knows he's in the best place now and in an environment which he likes and suits him.
"We want him to be a top player for Wales and they have the potential to play at the very highest level."
This season, Rodon has cemented his place under new manager Steve Cooper, starting another 20 Championship games in a campaign where the Swans have at times struggled defensively, but the centre-back's performances have still caught the eye.
What makes Manchester United interested?
Rodon's style has drawn comparisons with Manchester City defender John Stones from some quarters - and sports journalist Dominic Booth, who formerly covered Swansea for the South Wales Evening Post, told Sky Sports he was destined for bigger things.
"He brings the ball out from the back, he's comfortable with it at his feet. Rodon was part of the Swansea U23 side who won the Premier League cup in 2017, alongside the likes of Oli McBurnie and Dan James - it's been a good crop from the club's academy to break through.
"Rodon is a leader given his age, a bit of an old head on young shoulders. He's good in the air despite not being a huge presence, and looks destined for bigger things."