Beckford, who scored a hat-trick in the 4-0 win over Swindon at Wembley, handed the eight-year-old his shirt before cameras showed an unidentified woman wrestling it away from him.
A club statement read: “The club are fully aware of an incident in Sunday’s play-off final win over Swindon Town involving Jermaine Beckford’s match shirt.
“The club have been in consultation with the Football League and the parents of the child who is alleged to have had the shirt taken from him as seen on television cameras covering the match.
“Please note, no one involved in this situation is connected with Preston North End in any way, shape or form.
“The people seated in the area of the incident attended the Play-Off Final as part of the League One Football League Kids’ Cup Final which was played earlier in the day.
“We are delighted to say that Jermaine is more than happy to sign and present the child involved with his spare match shirt from the game. The club will also provide a stadium tour for the youngster.
“If the shirt that Jermaine wore is returned to us, we will of course present this to the child in question instead.
“The club will not be commenting further on the matter at this time and would like the focus returned to Sunday’s fantastic achievement and building towards what will be an exciting season in the Championship.”