The Pensions Regulator is unlikely to find strong legal grounds to demand Sir Philip Green put more money into the BHS pension scheme, an industry expert has warned.

“None of the documents that have been published on BHS show a smoking gun,” said John Ralfe, an independent pensions expert who acted as an adviser to MPs in their inquiry into the demise of BHS. “If the regulator decides it has a weak legal case it won’t pursue legal action.”

The regulator has powers to demand payments from employers and connected parties such as Green if it can show they have deliberately tried to avoid their obligations to a pension scheme or sold assets belonging to the scheme at less than their market value.

It is expected to release its report into the BHS scheme by the end of the year, as Green continues discussions about how he might meet his public commitment to “sort” the scheme, which had a deficit of as much as £571m at the time of the retailer’s collapse.

Those close to the discussions point out that the regulator has access to much more information than that in the public domain, much of which was published as part of a joint inquiry into the demise of BHS by the business innovations and skills committee and the work and pension select committee.

Ralfe says there is a considerable chance Green will be free to make an offer as he sees fit, in order to meet a moral obligation and repair his reputation, rather than being legally pursued by the regulator.

Green is under increasing pressure to come up with a deal before a House of Commons debate on Thursday about the MPs’ report into the demise of BHS and whether Green should be stripped of his knighthood.

He sold BHS for £1 in 2015 to investors led by Dominic Chappell, who had previously been declared bankrupt. The administrators were called in in April this year, later leading to the loss of 11,000 jobs and leaving a pensions black hole.

On Tuesday Green made a last-ditch plea to save his knighthood, hinting that he is close to agreeing a rescue deal for the BHS pension fund and stating he is “very, very, very sorry” for the collapse of the business.

In an interview with ITV News Green said his advisers were engaged in “strong dialogue” with the Pensions Regulator. “If we arrive at the place where we hope to arrive, there will be no requirement for [industry lifeboat] the Pension Protection Fund (PPF) [to step in],” he said. He would not put a figure on the amount he was prepared to invest in a rescue scheme but said he believed the situation was fixable.

The regulator later issued a statement saying it had “yet to receive a comprehensive and credible written proposal and have made clear what we require in this regard”. But it is understood that all sides are hoping to reach some kind of agreement before the year end.

Chris Martin, chair of the BHS pension fund’s board of trustees, told MPs at a new work and pensions committee inquiry into pensions regulation on Wednesday that discussions are continuing with Green. But he said there was no “single concrete proposal on the table at the moment”.

Martin told the committee that pension holders were not likely to get the full benefits they had expected before the collapse of BHS, although his aim was to secure a better deal than that offered under the PPF.

“I think the trustees have to be realistic about this, on behalf of the members,” he said, adding: “Settling somewhere between PPF and full benefits might be the best outcome that can be delivered.”

Ros Altmann, the former minister of state for pensions, also told the committee that she was “puzzled” as to why Green didn’t seek clearance from the Pensions Regulator before selling BHS in 2015.

“What I’ve not seen or understood was what plan there was for how the pensions deficit was going to be paid once the business had been sold,” Lady Altmann said.

Frank Field, chair of the work and pensions committee – which was one of two select committees that undertook the inquiry into BHS earlier this year, said on Wednesday that he believed there was little to justify Green holding on to his knighthood.

He told BBC Radio 4’s Today programme: “The key thing was coming up with a generous pension settlement, but he’s never done so. The flavour of this would change if he put concrete proposals which ensured that the pensioners were better off than they are now, heading for the Pension Protection Fund, that would help.”

The debate on Thursday will not automatically lead to a vote among MPs, but will be subject to the speaker’s approval. If MPs do vote to strip Green of his prestigious title, it would be “persuasive” only. The house’s honours forfeiture committee makes its decisions independently, although the prime minister could make a recommendation.

Field and Green have repeatedly clashed through the course of the parliamentary investigation into the high street retailer. And in a report concluding the investigation, MPs accused Green of “plundering” BHS, systematically taking money out of the business, leaving it on the brink of collapse and the pension fund in deficit.

In response, Green released a review by his lawyers saying the MPs’ report contained factual and legal errors. Green says he did everything possible to keep the business from going under.