Andretti, who runs several outfits in world motorsports including one of IndyCar's leading teams, has been working on an F1 bid for years and felt he finally had the missing ingredient to sway the sport by partnering with a top-level OEM in General Motors.
But most of F1's existing 10 teams - who have a say in any new team joining - are still hesitant to accept Andretti into the fold as an expansion team, as they would then have share prize money with an 11th team.
They also deem the current $200m dilution fund for new teams to be outdated and inadequate as F1's huge boom and its budget cap have now turned teams into valuable franchises.
With GM's support Andretti could well be positioned to deliver the added value that F1 craves, but it remains to be seen exactly how much the American manufacturer will be involved.
F1's cold shoulder has not gone down well with Andretti, who has publically criticised current teams for being "very greedy" and not looking out for what is best for the series.
It comes against a backdrop of F1 being perceived as elitist and not welcoming towards American involvement, despite expanding to three races in the country in 2023.
In an extended interview with Sky Sports's Martin Brundle, F1 CEO Domenicali states that F1 is not unwelcoming towards Andretti, but that his public criticism of current teams and his aggressive lobbying through the media and in the paddock was not smart.
PLUS: Why Andretti is both right and wrong with his F1 "greed" accusations
"We are very welcome to have anyone that is bringing value to the racing. That is not a problem of not welcoming, because that has been a wrong wording," Domenicali says.
"Andretti was very vocal about the will to enter in Formula 1. In my view, [it was] not smart to say that the teams are greedy to protect themselves, but that's my opinion, but there are others that are much less vocal that they would like to come into F1.
"I've discussed that with them very openly. And I said to them that I would act in a different way."
Domenicali says that like any other prospective entrants, Andretti will have to follow due process as its bid is carefully examined.
"We will make sure together with the FIA that the process will be respected and if all the elements are there, they will be very welcome," he insisted.
"And we don't have to overreact because someone is pushing the system. I believe that what is more vital is to protect the growth of the sport and also the sustainability of the teams that have invested in F1 in the times where this was different.
"The value of that investment today, from a purely commercial point of view, is much more different than it was just a couple of years ago.
"So, I think that the process will be done seriously, in the right way. And no one can put [up] with the anxiety to take the right decision, because someone is shouting.
"It seems that there is a personal negative attack on Andretti. That is not the case, we need to be serious and professional in evaluating all the elements."
He explained F1 is looking for a sustainable, long-term commitment to avoid getting burned by manufacturers suddenly dropping out or privateer expansion teams like Caterham, Manor and HRT coming and going on unstable footing.
"If you're able to join the family of Formula 1 you need to be really strong in terms of a sustainable plan for the future," he added.
"What has been one of the problems of the past few years is that we had so many teams coming in and coming out. And we need to make sure that we protect that for the future for the best of the sport. And that's it.
When asked by Brundle what the process of joining looks like, Domenicali said: "The tender is not out yet. It will be out soon because we have to decide together.
"And this will be out very, very soon, I think at the beginning of March."
