Sorrell, chief executive of WPP, posted an indicative 260p a share offer last week, but must decide whether to make a formal offer by the Takeover Panel's deadline of July 9.
Sorrell met some of the TNS shareholders at the end of last week, when hedge fund investors accounting for around 15% of the company encouraged him to press ahead.
Some analysts believe he could increase his bid to 270p, short of the 300p that would secure a recommendation from the TNS board, according to a report in the Guardian.
At the end of last week TNS rejected WPP's third takeover attempt, which valued the company at £1.08bn, and told it to "stop interfering".
TNS believes that a merger with market research company GfK is the best future for the company and on Friday it said that the board of GfK had voted in favour of the deal.
The planned merger of TNS and GfK would create the world's second-biggest market research company after AC Nielsen and would leave Kantar, WPP's research division, in fourth place, according to analysts.
Donald Brydon, chairman of TNS, dismissed WPP's offer last week saying that it significantly undervalued the company. He was quoted in the Wall Street Journal saying that Sorrell "wants to screw us up".
Brydon claimed that Sorrell's interest was motivated by a desire to frustrate its merger with GfK, which posed a serious competitive threat to Kantar.
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