The company said on Tuesday that the offer had been made after the close of trading on Monday.
The Affinity offer is the same as that made by the US private equity company Kohlberg Kravis Roberts on 7 June and at $3.50 a share prices Vocus as being worth $2.2 billion. Including debt, that comes to $3.3 billion.
Last week, Vocus opened its books to KKR to conduct due diligence in order to examine whether its offer could be taken further.
"The Board of Vocus now considers that the interests of shareholders will be best served by a formal process to thoroughly evaluate whether a change of control offer, at a price and on terms that the Board would recommend, can be secured," the company said in a statement.
"In order to ensure that the process operates as effectively as possible, the Board does not intend to make any further announcements unless and until a recommended offer is secured, or unless there is a development which it considers requires disclosure.
"The Vocus Board reiterates that there is no certainty that this process will result in an acceptable offer for Vocus, nor what the terms of any such offer would be, or whether there would be a recommendation by the Vocus Board.
"Vocus shareholders do not need to take any action in response at this time and the Vocus Board will update shareholders as appropriate."
In the last 10 months, Vocus has lost almost 70% of its value in its bid to incorporate a number of acquisitions, most recently the $807 million purchase of Nextgen.
Vocus provides telco, data, cloud and energy services through its Commander, Dodo and iPrimus brands. It also owns Nextgen Networks which it acquired in October 2016.
In May, Vocus was reported to be contemplating a sale of its data centres to service its debt. The company was said to have reached a preliminary agreement on the sale, with NEXTDC among those expressing an interest.
Credit Suisse and Goldman Sachs were reported to be around the sale.
Vocus has a fibre network across and in between major cities and 112 of 121 points of NBN interconnect.