The Gunners are reportedly targeting either Calvert-Lewin or Aston Villa striker Ollie Watkins, with current centre-forward Alexandre Lacazette likely to leave the club.
The Sun say boss Mikel Arteta is looking for a young goalscorer with Premier League experience, and Calvert-Lewin definitely ticks those boxes.
However, Mills claims that the England international should remain at Everton and, with Rafa Benitez as his manager, will 'thrive' when he returns from his injury.
"Right now, I wouldn't," Mills told talkSPORT, when asked if Calvert-Lewin should go to Arsenal.
"I think when you look at what Rafa Benitez has done, the two players he's brought in have really been beneficial.
"I think you look at his pedigree you look at Rafa Benitez's pedigree versus Mikel Arteta's pedigree - I know where I'm going, and I know how I believe in going to get it right.
"Long term, Everton have got obviously got a lot of money. They're very well-backed financially. They're possibly moving into a new stadium, sooner rather than later.
"I think Dominic Calvert-Lewin will thrive once he's back fit. He's been injured, don't forget, he's had this broken toe issue that's obviously kept him out.
"I think he will thrive under Rafa Benitez, and I think he will get even better. If he goes to Arsenal, I'm not sure.
"What's going to happen at Arsenal?"
He added: "Have they really got more chance of winning trophies? I would say Everton and Arsenal are probably on par at the moment, in terms of winning a trophy in the next five years.
"I don't think there's an awful lot between them."