Ole Gunnar Solskjaer wants to add proven quality to his side this summer after a dismal season which saw the Red Devils finish down in sixth in the Premier League.
Barcelona need to raise funds to sign Antoine Griezmann and Matthijs de Ligt and therefore could be tempted to sell.
Spanish outlet Sport claim United have made the approach for Rakitic with Ander Herrera set to leave the club.
The Croatian has won four La Liga titles and a Champions League at Barcelona.
The report claims Rakitic would demand a pay increase if he were to leave the Nou Camp for Old Trafford.
Barcelona want £50million for the midfielder and the 31-year-old could be tempted to leave due to the arrival of Frenkie de Jong from Ajax.
Manchester United boss Solskjaer is looking to bolster his midfield with Herrera leaving and the future of Paul Pogba uncertain.
But the Red Devils are also chasing De Ligt.
A move for Rakitic could help Barcelona complete a move for the 19-year-old.
And De Ligt revealed Cristiano Ronaldo told him to join Juventus on Sunday after Portugal beat Holland in the Nations League final.
“He asked me to come to Juventus,” he explained.
“I was a little shocked by that question, that’s why I laughed. I didn’t understand him at first.
“Just after the game you just feel bad that you’ve lost, that’s all you’re thinking about.”
De Ligt impressed this season in helping Ajax reach the Champions League semi-finals and win the Eredivisie title.
The centre-back underlines the main thing he will consider in leaving the Dutch champions is the playing time he will receive at his next club.
“Now I will go on vacation, I’ll think carefully about what is best for me and then I’ll decide,” he continued.
“For a young player like me, it’s important to go to a club where I can play consistently. That’s the main issue.”
United would love to add De Ligt to their ranks after a season of turmoil at Old Trafford saw them sack Jose Mourinho and hire Solskjaer.