Earlier, American Haas said the world's most famous female driver, who already races for his Nascar team, "would be the dream driver" for his 2016 outfit.
However, 32-year-old Patrick said this week that she is "happy where I am".
At the same time, there are rumblings that Haas, 61, and his new team principal Gunther Steiner are having fundamental disagreements about the project.
Steiner, for instance, is said to be opposed to involving Patrick.
"This is just the latest divergence that every day becomes more pronounced," said Omnicorse correspondent Franco Nugnes, claiming the disagreements could lead to Steiner's departure from Haas.
In the end, the team may opt for the more traditional 'middle of the grid' F1 model, featuring an experienced name alongside a so-called 'pay driver'.
Haas told F1 business journalist Christian Sylt: "Having an American driver would be one of those 'knock it out the ballpark'-type moments.
"I think initially what we really would like is probably a veteran formula one driver, preferably someone who has driven the 2014 car, to help us.
"Then the remaining seat would go to someone who has a lot of promise, that could be an American, or someone else that brings some sponsorship money to the table too," he added.