The Magpies are searching for their ninth manager in four years after parting company with Chris Kiwomya following Saturday's 1-0 home loss to Preston North End.
Kiwomya paid the price for a dismal start to the season which leaves Notts bottom of League One after losing 10 of their opening 13 games.
Danny Wilson is currently the bookmakers' favourite to take over at Meadow Lane, but Rodwell told Sky Sports News that they are still in the very early stages of searching for a replacement.
Rodwell said: "All we've done is have a long conversation with Chris, (assistant manager) Mick Jones and (coach) Andy Watson on Saturday night. We tried to do things in the right way.
"We broke the story at 10am on Sunday and one minute later the phone started going ballistic. We've started to compile names but we haven't spoken to anybody.
"We'll go through a process, talk to some people and see who's interested in the job, and take it from there."
County's development squad manager Steve Hodge will take charge of Tuesday's home game with Oldham and Rodwell is unsure whether they will have a new manager appointed in time for Saturday's game against Coventry.
"I don't know how realistic it is to have somebody in for Saturday," said Rodwell. "We've got a big game here against Oldham on Tuesday and Steve Hodge is taking charge of the team. He'll do an excellent job for us.
"I think the players are determined to turn it around. We're just concentrating on Oldham at the moment. We'll see where we are on Wednesday."
Rodwell revealed he spoke to the players following Kiwomya's departure and believes they can still have a 'very good season'.
"We've got good players in that dressing room," he stated. "We're not a bottom of the table team but results suggest that we are. We've got to change something and change it very quickly.
"We've still got faith in the players, but something clearly isn't quite right at the moment. They've got to take some responsibility, which they have accepted.
"We're bottom of the league but we don't intend on staying there. We've got a budget that is probably about eighth or ninth (highest) in the league. We don't want to be bottom of the division but there's a long way to go yet.
"I take great heart from statistics and last season Bournemouth had an awful start, but got promoted. I'm not suggested we will get promoted, just that we can still have a very good season."