Police are appealing for anyone who may have seen a black car being driven in a suspicious manner to get in touch, after several reports that a man in his 30s was approaching children.
Mid Ulster Councillor Denise Mullan said three concerned residents from the Windmill area of town told her about it earlier in the week, and that she advised them to call police.
Urging parents to be vigilant, Cllr Mullan said: “A woman told me about it on Monday morning, and then within the next few hours another lady told me the same thing.
“I heard it three or four times by lunch time.
“I would urge parents to speak to their children about stranger danger, so they know what to do if anyone approaches them.
“And if they do, please call the police.”
Cllr Mullan said she is not sure whether the incidents she heard about involved girls or boys - only that they were children.
The Irish News has reported that around six children were approached.
And police have confirmed that they are investigating one report in the Ballygawley Road area of town on Saturday, July 2.
It has been claimed that a man, described as being in his 30s, with black hair and a beard, stopped his car beside a group of children and offered one girl a lift.
He was also reported to be wearing a baseball cap, with a gold chain around his neck.
PSNI Inspector Neil Beck said: “I would appeal to anyone who may have seen a black car, possibly a Vauxhall Insignia, being driven in a suspicious manner in the area to contact police in Dungannon on 101 quoting reference 1350 02/07/16.”