There's speculation that Duke could boost his stake by underwriting a non-renounceable rights issue.
In that case he could, in theory, snap up new shares issued through a capital raising that are not taken up by other shareholders. Increasing his stake beyond 20 per cent would require a takeover bid or shareholder approval.
Duke, who owns almost 80 per cent of Briscoe Group, said it would not be "particularly clever of me to confirm or deny" the rumour.
"All I can tell you is that I really don't want to comment on speculation that's in the market," he said.
Pumpkin Patch chairman Peter Schuyt said there had not been any discussion about Duke increasing his stake "at the board level or with Rod".
As a possible takeover target, the retailer is certainly looking cheap.
Its share price has fallen almost 70 per cent in the past 12 months and it is trading at record lows, closing at 23c last night, valuing the company at $38.9 million.
Pumpkin Patch has been hit hard by online competition, rampant discounting and subdued consumer sentiment. It made a $10.2 million loss for the year to July, after a profit of $5.1 million a year earlier.
Briscoe Group - the owner of Rebel Sport, Briscoes and Living & Giving - has been a star performer among the listed retailers in recent years and Duke has a reputation for being a canny operator.
He said last year that Briscoe Group, which had a $60.2 million cash pile at the end of July, was eyeing possible acquisitions.
Paul Harrison, managing director at Salt Funds Management, which holds Pumpkin Patch shares, said a Duke takeover offer was a possibility.
But the childswear seller was a very different retail business to Briscoes or Rebel Sport, which largely sold "other people's products" rather than designing their own, he said.
Other major Pumpkin Patch shareholders include AMP and Kathmandu founder Jan Cameron.
