Sir Alex Ferguson appeared desperate for the Blackburn Rovers marksman to lead his attack alongside Eric Cantona and Andy Cole, but with Shearer's hometown club Newcastle United also in the frame, United had strong competition.
The emotional tie of a move to Tyneside would eventually prove too strong for Shearer, who signed on at St James' Park in a deal worth a record £15m.
So, which big name did United plump for to soften the blow? Gabriel Batistuta perhaps? Ronaldo or Patrick Kluivert? No, they instead shelled out just £1.5m to bring 23-year-old Ole Gunnar Solskjaer from Molde to Old Trafford.
It was a transfer that confused supporters and journalists alike. Having gone from chasing one of the most high profile centre-forwards in European football, United had landed a little-known Norwegian whose name required serious concentration to pronounce correctly.
Solskjaer had been tipped to play a bit-part role during his maiden season in the Premier League behind the more senior Cantona and Cole.
He didn't feature in the curtain-raising Charity Shield at Wembley, where United defeated Shearer's Newcastle 4-0, nor did he turn out against Wimbledon, in a match best remembered for David Beckham's halfway line exploits, on the opening day of the season. Ditto during the 2-2 draw with Everton.
However, he was named among the substitutes for United's second home clash of the campaign against Blackburn Rovers 17 years ago today. Another new Red Devil, Jordi Cruyff, opened the scoring, but visiting Rovers were soon in front thanks to strikes from Paul Warhurst and Lars Bohinen.
That goal from Solskjaer's countryman Bohinen prompted Ferguson to call for his new signing, who was thrown into the action in the 64th minute with the instruction of salvaging a share of the spoils for United.
He duly obliged just six minutes into his debut. Tim Flowers saved his initial shot, but as he would go on to do so often in a red shirt, Solskjaer fired in the rebound to secure a 2-2 draw for United.
In total, the soon-to-be-nicknamed 'Baby-faced Assassin' found the net on 18 occasions during the 1996-97 season.
The now-manager of Molde would very quickly become a hero among the United faithful, winning six Premier League titles, two FA Cups and a Champions League (thanks to his right boot) before a knee injury forced him to retire in 2007.