The Red Devils are keeping tabs on Elm, whose brother, David, used to play for Premier League rivals Fulham.
Many of the Old Trafford faithful have been clamouring for Sir Alex to bring in a creative midfielder, with Inter Milan star Wesley Sneijder having long been touted.
Whilst the Manchester United boss may see Tom Cleverley as the long-term solution, either way the reality is that the club are short on quality playmakers since Paul Scholes left.
The club don't seem to have many coming through the ranks either, especially with Paul Pogba's future up in the air.
And at the moment, the lack of attacking thrust through the middle is hurting Manchester United, which before the Wolverhampton game only managed to score more than two goals in a game back in October, when they played Aldershot in the League Cup.
Considering rivals Manchester City are well ahead in the scoring stakes, with the likes of Chelsea and Arsenal also gaining more attacking impetus, it seems like Manchester United could do with more goals from different positions, especially through the middle.
Of all Manchester United's current central midfielders, Anderson leads the goal-scoring charts with two goals, along with temporary holding midfielder Phil Jones. No other central midfielder has scored for Manchester United.
Should Manchester United Sign Rasmus Elm?
Overall that's a central midfield contribution of three goals and five assists.
So it's clear to see where Rasmus Elm comes in handy.
He impressed for Sweden against England at Wembley with his ability going forward, and has done the same for AZ Alkmaar in the Eredivisie this season.
In 16 league games so far, he's scored eight goals and made five assists, making a direct goalscoring contribution 81 percent of the matches he plays.
He's also a very consistent goal scorer when he does find the net; in a rich vein of form at the start of the season, he scored in six consecutive games for Alkmaar, with the team winning all of them.
It's clear to see Manchester United could do with some of that goal-scoring magic, seen as Elm has scored more than all of Manchester United's central midfielders combined, and made as many assists as them.
The Swede has also contributed well in the UEFA Europa League, scoring two goals and making five assists, with a direct goal-scoring contribution in 88 percent of the matches he plays.
It's evident to see how important it is to have goal-scoring players in the middle. Premier League leaders Manchester City have yielded nine goals and 18 assists out of their central midfield players.
Overall, with his attacking thrust, Rasmus Elm may well be the answer for Sir Alex Ferguson.
With goals and creativity in diminished supply at Old Trafford, his ability to provide exactly that makes him a logical solution for the Red Devils.
He may be no Wesley Sneijder or Mario Goetze, but for £6.5 million he could well provide as much value for money as them.