The BBC reports that Sir Philip is keen to reach a deal to resolve the collapsed department store's pension problems within weeks.
As a result, he has upped his offer to solve the funding shortfall from £250 million ($311 million) to £300 million.
Sir Philip Green bought BHS in 2000 for £200 million and his family took over £300 million in dividends out of the business shortly afterward. He sold it for £1 in 2015 and the department store collapsed into administration in April of this year. BHS had been struggling for years and a government report into its collapse blamed under-investment and excessive dividend payouts for sending it into a downward spiral.
BHS's 20,000 member pension scheme had an estimated funding shortfall of £275 million when the chain collapsed and that number has risen. MPs and the regulator have repeatedly called on Sir Philip, who is worth £4.1 billion according to Forbes, to meet that shortfall.
Sir Philip has said he would "sort" the problem but The Pensions Regulator (TPR) was reportedly unhappy with his offer of a plan that would involve a £250 million donation. TPR on Thursday launched an enforcement action against Sir Philip and Dominic Chappell, who bought the chain from Sir Philip.
The BBC reports that Sir Philip has now increased his offer in a bid to reach a quick solution to the problem. Enforcement actions often drag on for years in the courts.
Sir Philip's plan would involve buying out members of the pension scheme - giving them a lump sum now instead of accruing future payments. He says his plan would stop the scheme entering the so-called "Pensions lifeboat" - the state-backed Pensions Protection Fund (PPF), which gives a lower payout to retirees.
Sir Philip said in a statement on Thursday: "I have provided the Regulator with what I believe to be a credible and substantial proposal, with evidence and bank confirmation of cash availability, which would prevent the scheme from entering the Pension Protection Fund. This is in order to achieve a better outcome for the BHS pensioners."
MPs feel Sir Philip, who also owns chains like Topshop and Dorothy Perkins, is prevaricating. A non-binding motion to strip Sir Philip of his knighthood was unanimously supported in a recent House of Commons vote.