Pogba has been named in the PFA Premier League Team of the Year for 2018-19, having scored 13 goals and added nine assists.
But the France midfielder has also generated plenty of criticism for his inconsistent performances, with his future the subject of much debate.
Writing for The Telegraph, Carragher has opined that if a nine-figure offer arrives from Madrid, United should take the money and run.
"The immediate concern for Ole Gunnar Solskjaer is not who Manchester United should buy to rebuild the club. It is who they sell," Carragher said.
"There would be one name on the top of my list. Paul Pogba.
"If Real Madrid have £100 million and Zinedine Zidane wants him, United can solve the first of many problems as their long reconstruction gets underway."
Carragher continued by saying that Pogba has yet to live up to the £89m fee that United paid to bring him back from Juventus in 2016.
"I have heard Pogba described as world class," the 41-year-old continued.
"Being a World Cup winner does not automatically make you that. There is a difference between a world-class talent and a world-class player.
"He played his best football as a left midfielder in Turin without justifying what United paid to re-sign him, and he was a key member of France’s World Cup-winning side by enforcing a disciplined, tactical role rather than a spectacular or thrilling one.
"Even during those more encouraging moments he has shown nothing to suggest you can build a team around him. Juventus didn’t. France didn't."
With Pogba's deal nearing its expiration, Carragher has encouraged United to make a tough decision and sell the 26-year-old.
"United have a choice. Pogba has two years left on his current deal - with the club having an option for a third. If they do not sell Pogba they will have to offer him a new contract over the next year or two. What will he demand? Around £400,000 a week? Is he worth it?
"There has been enough damage done to United on the field since [Sir Alex] Ferguson’s retirement. The idea of them pandering to a player who has done nothing to warrant comparison with past and modern greats would cause even more damage to United’s reputation off the pitch."