Wilcox told BBC Lincolnshire he does not know why anyone would behave this way.
“I don’t want my reputation tarnished,” he said. “I’ve been in the game 40-plus years and I’ve never had anything like this before.”
Boles also issued a strong message to the person responsible for the hoax.
“Enough is enough,” he said. “Stop impersonating our manager, stop wasting the time of players and clubs across the non-league game and, quite frankly, get a life.”
Gainsborough, who reached the FA Cup second round in 2024-25 and first round last season under Wilcox, have made the Northern Premier League and Football Association aware of the incidents.
By making the issue public, Boles said he hoped the individual responsible would “grow up and go away”.
“It’s quite serious. If we were speaking directly to a player, it’s an illegal approach, it’s tapping up and the club can get in trouble with the FA and the leagues,” he told BBC Lincolnshire.
“More importantly for Russ, it’s his reputation that’s been dragged through the dirt.
“I don’t get what [anyone] would achieve out of it. Just pack it in.”
Wilcox has been with the club since 2023 following previous managerial spells at Scunthorpe United and York City.
He featured for Northampton Town, Hull City, Doncaster Rovers, Preston North End and Scunthorpe during his playing days.