Erik ten Hag is desperate to sign a striker and believes Hojlund is the most affordable option.
Talks have already taken place over an initial fee and are looking to wrap up a deal this summer, talkSPORT understands.
It comes after the Red Devils cooled their interest in Tottenham star Harry Kane after being told they would not sell to a Premier League rival.
United are eyeing Hojlund to join alongside another Serie A export in the shape of Andre Onana, who could arrive from Inter Milan this transfer window.
But talkSPORT understands that Ten Hag must sell players to bolster his funds with Dean Henderson, Fred and Anthony Elanga likely to leave in the coming weeks.
Read More On United
Man United players begin pre-season but Greenwood won't be there
Man United 'should have done more to keep Russo' amid fears they could lose Earps next
Hojlund has been earmarked as one of the most promising strikers in Europe.
At the age of 20, his goalscoring record so far has not been as jaw-dropping as someone like Erling Haaland.
But he has six goals in his last four caps for Denmark and scored 10 in all competitions for Atalanta last season.
Despite being 6ft1, the youngster has a surprisingly low level of gravity and has proved agile yet also has the strength to hold the ball up and lead the line.
Most read in Football
Everton to raid relegated Leeds and Chelsea to face awkward £100m situation
Rice and Timber's Arsenal medicals today, new Saliba contract, Man United push for Onana
I spent eight hours doing Spurs medical then joined rivals, one director threatened me
Henderson channels inner AJ as he throws punches in tough pre-season training workout
He also is deceptively quick and frequently latches on to long balls played over the top of defences, where he gets most of his goals.
However, one area he must work on is his finishing after missing 11 clear-cut chances last season in Serie A - the eighth-most in the division.
It saw Hojlund end the campaign with a conversion rate of just 16.7 per cent but the striker is undoubtedly raw and has talent that can be nurtured.
This is where attacking coach Benni McCarthy could prove vital, who has already worked wonders with Marcus Rashford.
The England international had his best season at Old Trafford after scoring 30 goals in all competitions while he owed much to the coaching of McCarthy.
Hojlund could explode into life much like Rashford at United under the right guidance but it is a gamble.
Considering the Red Devils were linked with Kane and Victor Osimhen, who scored 56 league goals between them last season, trusting a young forward like Hojlund is a shot in the dark.
However, having signed several stop-gap strikers such as Wout Weghorst and Odion Ighalo, it might be the right time to take a punt on a youngster.