Rugby Football Union chief executive Bill Sweeney says the organisation is looking to phase in cuts to funding for the Championship over two seasons.
Second-tier clubs say they are under threat after plans to slash their funding in the summer.
Currently the RFU pays the clubs about £534,000 per year, but that will drop to a 2015 level of £288,000 per club.
"We're looking at phasing that reduction over a longer period over two years," Sweeney told the BBC.
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Speaking for the first time about offering a potential compromise, he told Radio 4's Today: "We have gone to them and said 'look, we can reschedule these cuts over a period of two years, and work with you in terms of how to how to manage that situation'.
"The other thing we said is 'let's sit down and work together as a group and figure out what is the role of the Championship'.
"We haven't just cut this to save £3m. We've cut it because we are not clear of the benefit we're getting from that spending."
What have the clubs said?
When revealed last week, the cuts were described as "immoral and irresponsible" by Jersey Reds chairman Mark Morgan, while Bedford Blues chairman Geoff Irvine said the decision was "giving Premiership Rugby all that they want with regard to ring-fencing, in all but name and with none of the financial commitment or support".
A few days on, and while Nottingham chairman Alistair Bow has welcomed a phased introduction of the cuts, he says the decision still raises fundamental questions about the RFU's relationship with clubs outside the top tier.
"Putting the money to one side, that the RFU are still not clear on the benefits that they're getting from the Championship - on that basis, how can they value it?" he told BBC Sport.
"If they don't understand what the value is, how can they put monetary terms to it?
"It shows me even more that the RFU clearly are not in touch with rugby outside the Premiership."
He added: "If it's not a cost-saving exercise, they should continue with the funding until we have a strategy that we can either work to or can't work to.
"In December most clubs are looking their budgets for the following season and by the end of January your strategy for signing players is pretty much complete.
"At this time you'd expect your squad is pretty much sorted for the following two years, you're generally building for a two-year period, so looking at a two-year deal with the RFU is positive, but fundamentally it won't help move us forward."
Cornish Pirates, who are third in the Championship and are planning to build a new stadium with the aim of making the Premiership, will stay fully professional.
Pirates chairman Paul Durkin told BBC Sport: "It's still devaluing the Championship and we need to look and see what we can do with the RFU in the future to ensure the Championship is viable and seen as the second tier of professional rugby in England.
"I welcome the fact he's rethinking and looking at what needs to be done. It's a bit late as this first savage cut came when most clubs had already been involved in negotiating contracts for next season.
"He may have listened to what we said, but it's still a big issue for most clubs. Whatever that issue is, budgets had been looked at and had been worked on prior to any cuts.
"We as the Championship clubs need to get together to look at this, and discuss the implications over the two-year period. We need to see what the options are."
Coventry's executive chairman Jon Sharp helped put forward a blueprint for the future of the second tier before the funding cut was announced.
The plans, proposed with Ealing and Cornish Pirates, look at how the competition could be improved both on and off the field.
"The RFU's gesture is welcome, but it is regrettable that the clubs and our players, staff and supporters have had to go through a significant amount of angst to get to that point," Sharp said.
"There is certainly a lot to discuss, both between ourselves as Championship clubs and with the RFU as we look to take steps to make ourselves more self-sustaining, whether it is maximising our collective commercial rights, exploring potential new competitions, and making ourselves even more integral to the development of rugby in England."
'I do personally care very much'
The 12-team Championship was set up in 2009 as a professional second tier, replacing the 16-team National One, and saw its funding rise after England hosted the Rugby World Cup in 2015.
But since the Championship's inception it has been dominated on many occasions by the side relegated from the Premiership in the previous season. Only Exeter and London Welsh have ever won promotion having not been a Premiership shareholder.
Championship clubs are concerned that the cuts will make it easier for the Premiership to be ring-fenced, meaning there is no prospect of promotion to the top flight, or relegation from it.
But Sweeney says the £3m cut is needed if the RFU is to fund other parts of the game.
He continued: "We have 2,000 clubs out there. There are a lot of things that we want to do around developing the women's game, facilities and the grassroots game, and that needs funding and we simply can't fund everything to 100%.
"How do you apportion the money appropriately? And do we feel we're getting the right correct level of return?
"If you actually look at the spend on the Championship since 2015, our cuts in the professional game have gone down 15%, our cuts in the community game have gone down 5%. If you take out that 100% increase that Championship clubs (have had), it's gone down by 3%. So actually, they've borne less of the pain of the last five years."
Sweeney has also been accused of being "uncaring" by Bow - an accusation he denies.
"I can understand how he might feel that way but I can promise that I do personally care very much. We do care very much," said Sweeney.
"We've got 560 people here at the RFU. We're all very passionate about rugby. I've worked in the corporate sector now in the sports sector. And one of the things I can say to you is it's a privilege to work in the sport that you love.
"But when you make decisions in the sporting world, you're dealing with people's emotions, and it's very difficult."