In a video circulated on social media, the 37-year-old Portuguese forward is recorded limping off the pitch at Goodison Park before appearing to bend down and strike a mobile phone out of a young supporter's hands.
A statement issued by Merseyside Police said that they were liaising with both clubs regarding "reports of an assault" after the match.
"As players were leaving the pitch at 2.30pm it was reported that a boy was assaulted by one of the away team as they left the pitch," the statement read.
"Inquiries are underway and officers are currently working with Everton Football Club to review CCTV footage and are carrying out extensive witness enquiries to establish if an offense has taken place."
An update on the incident will be issued shortly, once further information has been gathered, a spokesperson for the Merseyside Police told CNN on Monday.
'It's never easy to deal with emotions in difficult moments'
Ronaldo apologized for his "outburst," offering to take the fan to a match at Old Trafford in a message posted on Instagram
The defeat against an Everton side fighting relegation dealt another significant blow to United's hopes of qualifying for the Champions League
next season, leaving them seventh in the Premier League and six points adrift of Tottenham Hotspur in fourth.
"It's never easy to deal with emotions in difficult moments such as the one we are facing," Ronaldo said.
"Nevertheless, we always have to be respectful, patient and set the example for all the youngsters who love the beautiful game.
"I would like to apologize for my outburst and, if possible, I would like to invite this supporter to watch a game at Old Trafford as a sign of fair-play and sportsmanship."
A spokesperson for Everton told CNN that the club were attempting to contact the boy's mother to invite the pair to the club's next home fixture, a league match against Leicester City on April 20.
The invitation was "to reassure the boy that incidents like that are not the norm at football matches and hopefully give him a positive memory of visits to Goodison Park," the spokesperson added.
Manchester United declined to comment on the incident, referring CNN to Ronaldo's apology posted on social media.