Ronaldo, 37, later apologised on social media.
Merseyside Police said they were investigating "reports of an alleged assault" at Goodison Park on Saturday.
A spokesperson said they were liaising with both Manchester United and Everton.
"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," they added.
"Inquiries are under way and officers are currently working with Everton to review CCTV footage and are carrying out extensive witness enquiries to establish if an offence has taken place.
"Anyone with information on this incident is asked to contact Merseyside Police social media desk via Twitter MerPolCC or Facebook Merseyside Police contact centre quoting reference number 228 of 9 April 2022."
Manchester United said they were aware of "an alleged incident" and would "co-operate with any police inquiries".
The result damaged United's hopes of qualifying for the Champions League and came against an Everton side who had been one point above the relegation zone before kick-off.
"It's never easy to deal with emotions in difficult moments such as the one we are facing," said Portugal international Ronaldo.
"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 apologise 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."
Photo PA Media Caption: Ronaldo appeared to knock a young fan's phone to the ground as he walked down the tunnel at Goodison