The Danish talisman has been linked with the Old Trafford side, who're rebuilding under new manager Erik ten Hag following a disastrous campaign.
Eriksen rose to stardom with his eye-catching performances with Tottenham Hotspur before joining Inter Milan, where he struggled with injuries.
A horrifying cardiac arrest at Euro 2020 last year almost ended his career, but a few months later, the 30-year-old returned to action with Brentford after terminating his contract with the Nerazzurri.
In 11 top-flight games, he scored once and assisted four times as Eriksen settled to life perfectly with the Bees. He is now wanted by top clubs again, including Manchester United.
The club reportedly scouted the star while he was on international duty with Denmark against Austria in the UEFA Nations League on Tuesday.
Eriksen once again played a huge role for his team by assisting Jens Stryger Larsen's 84th-minute winner as Denmark secured a 2-1 victory.
Given his talent and Premier League experience, Pallister feels Eriksen could succeed at Manchester United. Speaking to Lord Ping, he said:
"I think he's a fantastic player. I know he can do it in the Premier League, I saw him at Tottenham, he was outstanding there and that's what got him the big move.
"After all the problems he's had, it's great to see him back playing football. If he was to arrive at Manchester United, there's no doubt about him, he's proven it in the Premier League already that he can handle it."
Pallister further drew comparisons between Eriksen and Manchester United star Bruno Fernandes, whom he feels has a similar playing style to Eriksen.
"I think he's a little bit like Bruno Fernandes, I don't know if you'd be able to fit both of those players in the same team but it would be a nice combination to have I guess.
"He's got great quality, proven in the Premier League and on a free transfer, if you got him into your club, I'm sure every club in the world would be happy to have Eriksen in their side. It's one Ten Hag will maybe be looking at, it's not often that players of that class come up for free."
Manchester United in four-way race for Eriksen
Eriksen's contract with Brentford expires later this month, and several clubs are in the race for his signature.
According to reports, there's a 'four-way' race for the Danish midfielder between Manchester United, Everton, Tottenham and Brentford, who're keen to offer him a new deal.