Having kept a low-profile since being deposed from the chairmanship of Premier Oil in 1995, Mr Shaw now wants to raise cash for a series of projects in the Caspian Sea region. He has been chairman of Burren for less than a year.
A private placement is currently being organised by ING Barings and Finian O'Sullivan, Burren's chief executive, confirmed that it would be followed by a stock market launch.
He said: "We are working at an Initial Public Offering for Burren before the end of this year. We have been doing detailed work with Barings on the value of the company."
Outsiders said Burren could be worth more than pounds 80m on the basis of very encouraging flow rates from wells that are already producing in the Nebit Dag region of Turkmenistan.
If the IPO takes off, it will offer a triumphant return for Mr Shaw with the Caspian area representing all the frontier characteristics of the UK North Sea in the early 1970s.
Mr Shaw, who still has a small personal holding in Premier, left the UK explorer after holding the chairmanship there for 24 years. He was removed from his post against his will.
Over that period Mr Shaw, now aged 76, raised the asset value of Premier from pounds 175,000 to pounds 130m through a combination of exploration success and takeover.
But it was not all good times. Analysts, never favourites with Mr Shaw, remember Premier's share price hitting 90p but also dipping to 17p. Yesterday it was stable at 44.5p.
A larger-than-life-American who was a former diplomat and then bomber pilot, Mr Shaw became resident in Britain 50 years ago but still carries a heavy east coast drawl.
When he bid farewell to Premier, he said he had no intention of retiring. He instantly found directorships with Australia's Eastern Petroleum and the trading company, Maxoil.
He was talked into joining Burren by Mr O'Sullivan, a geophysicist. Burren's key asset is a 25 per cent interest in five Turkmenistan oil and gas fields which make up the Nebit Dag interest. Burren is working there alongside operator Mobil (40 per cent) and Monument Oil and Gas (35 per cent).
Last week Monument told analysts that latest reports from the area were extremely encouraging and said there was the potential to double the value of Monument's holdings.
Monument rushed out new figures on reserves to head off their publication in the financing documents of Burren. As well as Nebit Dag, Burren has interests in two oil and gas condensate fields in Saratov Province of Russia.
Burren has also established a shipping arm, VCTT, which is already has a contract to charter local tankers and is carrying oil from the Caspian area via rivers and canals.