Known as the Poynton Consortium, the group comprises former Alinta chairman John Poynton and executives Chris Indermaur, Stephen Pearce and Murray King.
The consortium, which unsuccessfully tried to take control of Alinta earlier this year, started buying up Tap Oil shares several months ago and now holds about 9% of the stock.
Tap’s outgoing chief executive Paul Underwood told AAP he was recently approached by the group and asked to put one of its representatives on the board.
The news service said Underwood challenged the group to either launch a bid or let the board do its job.