Stoddart, an Australian with strong local support in his hometown, wants to race last year's car with unmodified aerodynamics which do not conform to new regulations.
To do so he needs the agreement of all the other teams and he told Reuters on Thursday that Todt's signature was the only one still missing.
Stoddart said he was confident the Formula One world champions' would give their permission, despite Todt's previous refusal. Ferrari are also starting the season with their 2004 car, but with revised aerodynamics.
"Tonight when Todt gets in I'm going to go to his hotel," the Australian said.
"I hope that this can all be sorted out in the hotel away from the track and that tomorrow morning I can proudly say there are 10 signatures.
"Effectively without that signature we're in trouble.
"They are keeping me out but let's hope that is over."
Stoddart said Minardi, who scored just one point last year and are regularly several seconds off race pace, had passed the official International Automobile Federation (FIA) safety scrutineering.
"The FIA side is finished. The car complies with the 2005 safety regulations," he said.
"Remember (FIA president) Max (Mosley) saying that if the cars had this bodywork that they wouldn't be given their scrutineering tags. Well, the FIA's pulled away from that."
BURNING MERCHANDISE
Minardi do have their 2005 aerodynamics package in Melbourne but have not tested it, raising potential safety concerns. They plan to introduce their new car next month.
Todt was due to arrive in Melbourne on Thursday evening.
"I am actually (confident), I'm not complacent about it but I am because I don't think at the end of the day even Todt could take this any further," Stoddart said when asked about the likely outcome of their meeting.
"Already there's people threatening to burn Ferrari merchandise outside and everything.
"I believed he was going to sign. I rang him up and he said 'I'm not going to sign because you have had bad things to say about Ferrari for the last three months,'" added the Australian of their previous discussion.
"Then he said 'I've got to go' and put the phone down."
Australian Mark Webber, who finished fifth on his grand prix debut with Minardi in 2002 and now drives for Williams, said he would be very disappointed if the team could not compete.
"The fans are very much behind the Minardi team, they're basically a family team," he said.
Ferrari's seven times world champion Michael Schumacher said he also had a soft spot for Italy's 'other' team, but questioned whether their position was correct.
"Imagine if they made a new rule in soccer where you start playing with 10 people and there's always one team that's not doing very well," he said.
"Would you accept that they will be allowed to play with 11?"