The Norway international has attracted interest from numerous Premier League teams, despite enduring an injury-plagued campaign for the relegated Blades.
Reports from France earlier in the week suggested that Arsenal had stolen a march on their domestic rivals by tabling a £17.2m bid for Berge.
However, speaking ahead of Wednesday's trip to Newcastle United in their penultimate game of the season, Heckingbottom insisted that is not the case.
"No, no offers. We know how things work," he told reporters.
"If there is anything happening for our players it's bubbling away behind the scenes until windows open and things can possibly happen, but it is all paper talk.
"Most of the questions you get asked about possible targets never ever comes to fruition.
"Ninety-odd percent of it will never surface. At the minute it is all just rumour and speculation. We've had no bids for our players."
Berge has started just 13 Premier League games this season and will play no part in United's final two games against Newcastle and Burnley.
Everton are also believed to be in a strong position to sign the 23-year-old when the transfer window reopens next month.