Kessie has been in fine form for Atalanta this season, helping them up to fifth in the Serie A table.
The Sun claim the 20-year-old has been recommended to Jose Mourinho by fellow Ivory Coast international Bailly.
It is claimed scouts watched him in action at the Africa Cup of Nations and were impressed with what they saw.
A source opened up on the deal saying: “Bailly has flagged his international pal Kessie to the club. He thinks he could be the new Yaya.
“He was watched during the AFCON and they like what they see. There is other clubs looking at him as well.”
Manchester United player ratings against Saint-Etienne in Europa League
Thu, February 16, 2017
Express Sport brings you Manchester United's player ratings against Saint-Etienne in the Europa League
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Express Sport brings you Manchester United player ratings against Saint Etienne
Franck Kessie has reportedly been suggested to Manchester United
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Anthony Martial has been backed to hit top form for Manchester United
Paul Scholes thinks Anthony Martial isn’t cut out to be a striker.
But the United legend reckons the France international has the capacity to score 25 goals a season from out wide.
"We've all seen he's capable of scoring goals, beating players – but we're not seeing it enough," Scholes said.
"He's still a young player, we have to remember that. I think when you look at wide players, they're always the most inconsistent on the pitch, anyway. I think that will come with him.
"And if that does come and the confidence really comes out of him, I think he's got 25 goals a season in him."
Martial has scored just seven goals this season but has recently broken back into Mourinho’s side.
Ibrahimovic has scored the same amount of hat-tricks for Manchester United as Ronaldo
Zlatan Ibrahimovic has now scored as many United hat-tricks as Cristiano Ronaldo.
Ronaldo managed just a single treble during a Premier League match against Newcastle in 2008.
That was in his fifth season at Old Trafford – but Ibrahimovic has managed it in his first.
The 35-year-old achieved the feat during United’s 3-0 Europa League last-32 win over Saint-Etienne last night.
It also brought his tally for the season up to 23.