The fee offered by the Merseysiders for the 17-year-old Scotland youth international is believed to be lower than the £250,000 reported which has angered the Northern Irishman.
Shiels also confirmed defender Jeroen Tesselaar, who went off with a hamstring injury in Tuesday's 2-1 Scottish Communities League Cup defeat by Stenhousemuir at Rugby Park, would be "out for six weeks".
But on the prospect of losing one of his bright young prospects for good, he said: "There are all sorts of rumours going around but at this moment he is playing for me on Saturday. They (Everton) have made a derisory offer which was disrespectful to the club and unsettled the player which I wasn't happy about. But that can change as you know."
Shiels was surprised with recent reports linking Rangers with his keeper Cammy Bell.
"I find that hard to believe," he said. "It is a bit late, isn't it? There's not been a thing (from Rangers). If they were wanting him you would have thought it would have happened maybe a week ago but I can't see it happening now."