Beaumont hopes the deal will save the future of the Heineken Cup.
He will meet the 12 English clubs next Wednesday and is prepared to use his influence by acting as a mediator in a row which threatens to wreck Europe's premier competition.
The Premiership and French clubs have served notice on the current European participation agreement, seeking changes to the qualification system and greater financial rewards from 2016.
Premiership Rugby agreed a £152million contract with BT Vision, with the aim of establishing a "dazzling new European tournament" when the current accord ends.
But the PRO12 clubs have so far rejected Premiership Rugby's proposals and if there is no agreement within the next six months that could be the end of the Heineken Cup.
Beaumont brokered a peace deal when the RFU were involved in a television rights war in 1999 and is ready to step into the breach for the English clubs.
"As chairman of the RFU and being involved in the Six Nations we have a meeting coming up in Rome and this (the current European row) will be an agenda item," Beaumont said.
"I would like to think with the contacts that I have got and the relationships I have built up amongst our neighbours I will try and use my best endeavours. That is what the RFU wants.
"We are keen to resolve this situation. We don't want to be at loggerheads with our neighbours all the time or the clubs. Why would we?"
Beaumont is looking to help Premiership Rugby despite being told that the English clubs broke their contract with the RFU in signing the BT deal because it was not sanctioned by Twickenham.
"I am not a lawyer. I am advised that there could well have been a breach in the contract, but at the moment let's keep the door open, let's not take sides," Beaumont said.
"Let's keep talking and we have still got a fair bit to go.
"We have got to try and get on and reach an amicable settlement which suits all parties."
In 1999, Beaumont brokered a deal that saw England reinstated into the Five Nations after they had been expelled for signing their own broadcast deal with Sky.
Beaumont drove to the Drum and Monkey pub in Glasgow to meet Five Nations chairman Allan Hosie and thrash out the agreement over a pie and a pint.
The situation facing European club rugby is even more complicated.
The Rabodirect PRO12 teams currently pocket 52% of the income from European tournaments, with Premiership Rugby and their French counterparts receiving 24%.
The English and French clubs want that split to be 33% each but Premiership Rugby's plan, backed by BT, would also leave the Rabodirect clubs better off.
However, the Celtic nations will not accept plans from the English and French clubs to reduce the number of teams in the Heineken Cup and force all PRO12 teams to qualify on merit.
That is the key sticking point in the negotiations and there was no progress made in the first round of talks held in Dublin on Tuesday.
A further meeting is now scheduled between the key stakeholders for October 8 in Rome.
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