The news comes following remarks made by Robert Lerwill, chief executive of Aegis, saying that he wanted his WPP counterpart, Sir Martin Sorrell, to make a decision "sooner not later" on whether or not they will bid.
The Times says today that Aegis wants the potential bidders to "put up or shut up", although there have been disagreements about when the final deadline should be.
Aegis refused to comment on the report, which says that a deadline of late November could be imposed shortly.
In his interview with The Times, Lerwill says that in a meeting with Vincent Bollore, the French businessman who now owns 20.8% of Aegis, there was no suggestion of a link with Media Planning Group-owning Havas, the company of which Bollore is chairman.
Bollore has continued to buy shares in Aegis, even since rival French advertising group Publicis Groupe said that it would not be making an offer for the company.
Aegis owns the media networks Carat and Vizeum, along with market research company Synovate, and the digital network Isobar.
Sorrell confirmed earlier this month that WPP is interested in acquiring Synovate and would break the company up if it did acquire it.
The WPP chief described Aegis's research arm as "interesting" and said that it had always sat "rather uncomfortably with the media buying business".
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