Ex-Galway forward and selector Brendan Lynskey and U21 manager Johnny Ryan had been nominated for the role and, like Cunningham and Kenny, were interviewed for the position last Wednesday.
Both Cunningham and Kenny are expected to be interviewed again this week ahead of a recommendation being made to the board on September 29.
Cunningham is seeking an extension to his three-year term in charge, while Kenny hopes to succeed the St Thomas man he worked with for two seasons before stepping down last year.
Kenny is currently in charge of Dublin senior club side Cuala and is believed to have minor manager Mattie Murphy included as part of his prospective set-up.
The selection committee comprises former Galway manager Conor Hayes, county chairman Noel Treacy, chief executive John Hynes, treasurer Bernie O’Connor and from the hurling board chairman Joe Byrne, secretary Pat Kearney and treasurer John McCann.
Galway’s season ended in disappointment with the qualifier defeat to Tipperary, a week after they exited the Leinster championship losing to Kilkenny in a semi-final replay. However, Cunningham insisted the county were in a good place. “The performance was there, they’re thriving really. In particular we had five or six new players this year who hadn’t played championship before. We’re in a building process, happy with that though not happy with the result — we were very, very close.”
Two-time All-Ireland winner Kevin Walsh remains favourite to take over from Alan Mulholland as Galway senior football boss. The former Sligo manager has been nominated along with ex-Galway selector Pete Warren.
Meanwhile, top trainer Paddy Tally has joined Carrickmore’s coaching team ahead of the Tyrone SFC final later this month.
His arrival at Pairc Colmcille is a major boost to the club’s bid to land a record 16th title with victory over Omagh on September 28.
Tally trained the 2003 All-Ireland-winning Tyrone team, as well as the county’s U21 side. He later worked with successive Down managers Ross Carr and James McCartan, and for the past two seasons he has been Derry trainer, under manager Brian McIver.
“We have had Paddy in a few nights, and he has been very good, and the players seem to be enjoying it,” said Carrickmore manager Brian Gormley.