MacKenzie is believed to have made the move in order to secure investment for a strategy of expanding The Wireless Group's talk radio portfolio.
Existing shareholders are not believed to be supportive of such a strategy, so MacKenzie, who owns 6% of the company, has turned to private equity.
His backer is believed to be Veronis Suhler Stevenson, a US media investment specialist.
According to The Sunday Times, MacKenzie's offer of 90p per share was accepted by the company's non-executive directors on Friday, putting the deal in the region of £100m.
News International, which owns 30%, and Liberty Media, which owns 28%, did not comment on the speculation. It is expected that the company will formally contact its shareholders this week.
The Wireless Group owns national station TalkSPORT as well as 16 local radio stations. After winning the Edinburgh FM licence in December, it is due to begin broadcasting there later this year and has also submitted a bid for the Manchester FM licence.
News of the deal comes just after it was revealed that Scottish Media Group has rejected a £100m bid for Virgin Radio from Lord Waheed Alli, who was backed by private equity firm 3i.
Investors are waiting to see if Chrysalis, which recently bid £100m for Guardian Media Group's radio assets, will return with an improved offer.
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