The existence of the offer was revealed by TMN's independent non-executive directors in a statement to the stockmarket today, made without the consent of August Equity.
Mark Smith, chief executive, and Craig Dixon, chief financial officer, are working with private equity company August Equity.
TMN's share price has risen 28% from Friday's close to hit 55.4p at midday today.
In March, TMN rejected a £40m cash and shares offer from Tangent Communications, the marketing services group backed by former Carlton TV chairman Michael Green, saying it "substantially undervalued" the company.
The non-executive directors formed a committee to evaluate offers for the company and gave Smith and Dixon permission to pursue their interest.
There were reports of interest from other suitors including Trinity Mirror, before Tangent withdrew its offer in early April.
One shareholder, Mike Danson, has built a big stake in TMN since Tangent made its offer. That was when he emerged on the shareholder register with a 7% holding, which he has now increased to 24%.
Danson is the former chief executive and founder of Datamonitor, which was sold to Informa last year.