The 'Blowin' in the Wind' singer recently said he would be "excited" to collaborate with the former Beatles star, and after Paul also indicated his interest in a duet, they are planning to work on material in the US.
A source told Britain's Daily Express newspaper: "Paul has a home in California not too far from Bob's so the idea is for the two to meet when Paul is in California over the summer.
"They'll obviously want to work together in privacy and it is felt one of their homes would be the best option.
"Getting these two together for any length of time is difficult schedule wise but they both want this to work out. The music that comes out of this will be fascinating to hear."
It would not be the first time 68-year-old Bob has teamed up with a former Beatle.
He was a member of supergroup the Traveling Wilburys with the late George Harrison in the 1980s.
Meanwhile, Paul, 66, is reportedly livid he will have to pay out hundreds of pounds on a new TV following the switch from analogue to digital programming in the UK.
A source told MSNBC: "He's furious over this switch to digital cable. He doesn't think it's right that you have to either go to the trouble of getting an adapter or you have to buy a new TV, which he should do anyway.
"You'd think he'd have an apartment full of flat screen TVs but really, he's got these old clunky sets in this tiny New York apartment."