The Real Housewives of Beverly Hills star suggested the idea to McCain on Wednesday's episode of Watch What Happens Live With Andy Cohen. "Could you be the vice presidential nominee?!" she excitedly said as McCain discussed how much she admires the politician.
"I don't know who it is!" McCain replied.
"He said he was going to pick a woman," Jayne noted. "It should be you!"
"Probably more like Kamala [Harris] or someone like that," McCain said. "I have no idea."
"I'm not friends withKamala," Jayne cracked.
The pair started their conversation about Biden -- the presumptive Democratic presidential nominee -- as Cohen asked whether McCain, a Republican, would be voting for Biden over President Donald Trump in the next election.
"I just had a long conversation with him on Saturday. I love him dearly," McCain said of Biden. "I keep telling everyone, 'I will promise you, you will know who I'm voting for.' But it won't take a rocket scientist to know there is one who has made pain in my life a living hell and one who shepherded me through the grief process. Shouldn't be rocket science."
Cohen followed up, asking whether Biden could be the first Democrat the McCain family has ever voted for. The View co-host's father, late Arizona Senator John McCain, was a Republican. He ran against Barack Obama in the 2008 presidential election, but was great friends with Biden.
"It's really tough because, again, my parents met and ended up getting married in part because of [Biden and his wife, Jill]. They got drunk and danced together. They're old, old friends," McCain said. "He's been so integral in my life, especially since my dad got sick."
"I'm always like, my heart over my head in so many different ways, and the Trumps, they're always making my mom cry. I just think politics is personal too and character is really important. And [I want] someone who's going to tamp down... fear and anger, instead of making it worse."
During her appearance on Watch What Happens Live, McCain also discussed her pregnancy -- sharing that she's trying to be "excited" amid all else going on with the coronavirus pandemic -- and opened up about her strong reaction to Elisabeth Hasselbeck's appearance on The View.
Hasselbeck downplayed the severity of the coronavirus on the talk show, saying that she would "pray" away the pandemic.
"They sent me a screen shot of my face when she said that. I took the virus seriously from the very beginning and thought this rhetoric was dangerous. I think it's really, really unfortunate and dangerous that she said that. I don't need to co-host with her again," McCain told Cohen.
"It's unfortunate because I've been a huge fan for a long time," she added. "Anybody who is screwing around with this virus and putting out misinformation, I don't really have a lot of time for right now."