The Takeover Panel has given the Dutch chemicals company Akzo Nobel until 9 August to table a takeover offer for UK rival ICI.
Akzo Nobel made a £7.2bn takeover proposal on 4 June that ICI rejected, saying it undervalued the company.
The Financial Times says Akzo managers have been sounding out ICI shareholders about a further offer though there has been no formal contact with the firm.
ICI has seen its share price jump 40% in the past six months.
Helping drive the share price higher has been persistent speculation that ICI was a takeover target.
Akzo Nobel has plenty of cash to spend on acquisitions, having sold its pharmaceuticals division to Schering-Plough for £11bn in cash in March.
However, one potential fly in the ointment is that Akzo already owns Crown, which has 14% of the UK paint market.
ICI owns Dulux, which has 40% of the market, and analysts said that a takeover by Akzo may raise competition concerns.
One way around competition concerns would be if Akzo sold Crown UK, analysts said.