The singer, who is in the midst of a divorce battle with his estranged wife Heather Mills, was linked to New York businesswoman Nancy Shevell just a few weeks ago.
However, according to Britain's News of the World, McCartney was more recently seen strolling the grounds of a stately home with the thrice-divorced Arquette.
The "Pulp Fiction" and "Desperately Seeking Susan" star has also been seen at McCartney's house.
Asked about reports linking McCartney and Arquette, Shevell told the New York Daily News, "I'm sorry, I don't want to talk about it," then turned around and went back inside her New York apartment building.
When she emerged again later, she told reporters waiting by her door, "I'm sorry I have nothing to say, but it's OK, you're not bothering me."