Lancaster guided his new-look England side to second place in the RBS 6 Nations championship with four wins from five matches.
Rugby Football Union chief executive Ian Ritchie is expected to begin interviewing shortlisted candidates for the permanent job in the coming week.
Mallett, the former South Africa and Italy coach, was first contacted by the RFU in December and is thought to have applied along with Lancaster.
Speaking to Supersport television in South Africa, Mallett said: "You can't see the RFU wanting to change a coach when you look at the team and how happy they are."
Lancaster took the reins on a temporary basis after Martin Johnson stood down in the wake of England's World Cup debacle and he began to build a new team.
The 42-year-old would not comment specifically on the recruitment process, but his application has grown in strength week on week through the campaign.
England beat Scotland at Murrayfield for the first time in eight years, they ended France's 10-match unbeaten run in Paris and today's [Saturday] victory was only England's second against Ireland in nine championship meetings.
"I've just got to trust in the process. I've been on the other side of the fence interviewing people and I understand what people need to do," Lancaster said.
"I've got no problems with that. I will just enjoy the moment. I don't know (when I will be told).
"Obviously, there's a bit made of the head coaching job, but I've got two lieutenants either side of me who deserve a bit of credit as well."
England rounded off the Six Nations with a stunning demolition job on Ireland led by a ferocious scrummaging performance, which earned a penalty try and allowed Owen Farrell to kick 20 points.
Scrum-half Ben Youngs came on to add a late second try with a quick tap penalty, which had been earned by another destructive England set piece effort.
"First of all, I just want to express my pride in the performance today [Saturday]. We talked a lot as a group about team belief, believing in each other and also in self-belief," Lancaster said.
"The team came together eight weeks ago at a Yorkshire Two club in Leeds. Obviously it's been a great journey.
"The second half was outstanding, against an Ireland side that has been together a long time and are clearly well-coached.
"I'm sure the English fans enjoyed the scrummaging-fest and the control we put on the game. I'm delighted really and it's been a great week for me.
"I'm delighted for the players, the management, who've really bought into this new England team and the supporters.
"The campaign has probably exceeded most people's expectations, but I think, ourselves, we've always known we've had a good group of players and we've always believed in our ability to get the best out of them."
England forwards coach Graham Rowntree, who was part of the World Cup campaign, paid tribute to the rebuilding job Lancaster has done with the national team.
"We've been born again as a new team under Stuart. It's hard to reflect on that (the World Cup). That was such a long time ago and we've come such a long way," Rowntree said.
"It's been an emotional ride this last eight weeks but the credit tonight lies completely with that young group of players.
"I look at the pack today (Saturday) and half that pack debuted in Scotland. A new set of forwards, a front row average age 24 and they've driven everything.
"I've kept an eye on what they've done but their passion and the energy and their want to do well for each other, I'm so proud of them.
"For me that makes the World Cup a distant memory. We are born again, made everyone proud of us again and we are still not the finished article. There's still loads more to come."