Hill is in the middle of a two-season contract as player/manager of Ramblers, Richardson lives in Dublin and is not actively involved in football. Both have strong connections to the Cork club.
Chairman Brian Lennox stressed yesterday he had no preconceived ideas as to who might be appointed to the position vacated by Murphy.
"We will advertise for applications on Wednesday in the English and Irish press and the competition is open to everybody," he said.
Public opinion immediately registered Richardson and Hill as two potential appointees but, in Hill's case, Cork would have to secure Cobh's co-operation if they were to interview him.
Hill joined Cobh at the start of the season after winning a League Championship and FAI Cup medals with Bohemians. Prior to that he spent several successful seasons with Cork City.
He said yesterday: "Of course you would have to be interested in the position, but right now I am manager of Cobh and totally committed to the cause of the club.
"We worked hard to secure a play-off against Drogheda on Wednesday and that is the only thing I am interested in now. I want to lead them into the Premier Division."
When pressed on the Cork situation, he said: "I am under contract and will not be applying for any job while I am committed to Cobh.
"If Cork are interested they will have to approach the club and if they do that then, of course, I will talk to them. But if there is anything to be done then it will all be above board."
Richardson has been out of football since he parted with Shamrock Rovers at the start of the season. He spent two seasons with Cork City during difficult times when they were trying to get established in the new stadium in Bishopstown. "I enjoyed my time in Cork tremendously but I left because of a less-than-professional
approach within the club. I felt the progress of the club was impeded because there was no real ambition in the club," he said.
Since then the club has gone through two changes of ownership and Richardson said: "I have always said the two best jobs in the league were at Cork City and Shamrock Rovers. The fact the Cork job is now being made a full-time post is interesting, but I have a few other irons in the fire."
The man behind the drive to progress Cork City is Brian Lennox, who took over control of the club 12 months ago. He stressed that he was determined to lead the club into a more professional era clearly following the pattern laid down by successful Dublin clubs Bohemians and Shelbourne.
Murphy, who is vacating the job as manager to take up a newly created position as technical director, pointed to the path of development he and the chairman have mapped out for the club.
"We need four under-age teams. We have U21, youths, U17 and we need an U16 team as well. If you want to qualify and compete in Europe on a regular basis there is a lot of work to be done, we just don't have enough bodies in the club.
"A big investment is needed in youth and I think Cork City will, in the end, have a successful academy with players improving constantly and stepping up gradually so we will have players emerging who are ready to come into the first team all the time," he said.
"These would be contracted players intent on making a career in professional football because we have to increase our number of full-time professional players. This is going to cost the club money but the chairman is committed to that.
"This year we will be looking to bring in three or four more on professional contracts with more to follow the following year. In two or three years time, hopefully, we'll have a full-time club."