A source said officials at the NIO had been impressed with his work in dealing with other marching controversies.
"He is someone with a proven track record," the source said.
It is understood that no approach has been made as yet to Mr Roddy.
When contacted yesterday, he told the Belfast Telegraph: "I know nothing about this at all."
And a senior Orange Order source said it was unaware of any approach to Mr Roddy in the long-standing impasse over a parade along the mainly nationalist Garvaghy Road in the Co Armagh town.
The commission, however, is said to want Mr Roddy to be involved in efforts to try and initiate discussions between Portadown District Lodge No.1 and the Garvaghy Road Residents Coalition.
There have been suggestions that the agreement that ended the Ardoyne dispute last weekend could be a template for Drumcree.
The contentious Ardoyne march went ahead early last Saturday following an agreement between Orange lodges and the nationalist Crumlin Ardoyne Residents Association (Cara) that Secretary of State James Brokenshire said provided a "platform of co-operation" which could be built on.
Former Derry city centre manager Mr Roddy and ex-Methodist Church president the Reverend Harold Good were involved in brokering the deal to end the long-running Ardoyne dispute.
Mr Roddy, who was awarded an MBE earlier this year for services to business and the community, was also at the forefront of successful negotiations between the two communities over parading in the Maiden City.
Policing the Drumcree dispute costs the PSNI more than £23,000 a year, as Orange Order members still stage a token protest each week.
Every Sunday a number of lodge members walk from Drumcree Church of Ireland to a nearby small bridge, where they hand in a letter of protest to police.
It means three PSNI officers attending for around two hours most weeks.
And then on the day of the once-annual parade itself - the Sunday before July 12 - around 60 PSNI officers are deployed for several hours.
However, at the height of the Drumcree dispute during the mid-1990s the annual cost of the policing bill was estimated to be £14m.
Breandan MacCionnaith, who for years was the main spokesman for the Garvaghy Road Residents Coalition, has said the issue is a "dead duck" and no longer on the radar.
Earlier this year Portadown district master Darryl Hewitt revealed - after years of the Orange Order refusing to talk to residents - attempts had been made in 2014 to engage with them through a Catholic priest.
But the proposal was rejected out-of-hand by the Garvaghy Road residents.
And Mr Hewitt claimed a Parades Commission member had said he had "not heard anything new coming out of Portadown" for three years.