The former Beatle - worth an estimated $1,610 million - instructed lawyers to make the offer following days of "intense and heightened" talks.
This is said to be the first time Paul, eager to avoid a public court battle, has agreed to open negotiations.
A source told Britain's Daily Mirror newspaper: "This is the closest we have been to an arrangement. Paul is absolutely determined to keep his private life hidden from public scrutiny.
"The divorce has now been dragging on for nine months. There is war weariness creeping in."
"Lawyers have been in feverish discussions for the past ten days, but isn't simply a matter of him making an offer and it being accepted."
Paul, 64, reportedly sanctioned the offer after Heather's sister Fiona accused him of putting his three-year-old daughter Beatrice at risk by refusing to pay for security for Heather.
The source added: "They are almost there. The channels of communication have opened up."
"It's the first time a deal has been mentioned and both sides hope a settlement will be reached. Unless there is a major hiccup, they will not be going to court."
Terms will be drawn up by Anthony Julius and Fiona Shackleton - the same lawyers who dealt with Prince Charles and Princess Diana's divorce - and they are hoping to settle the deal within a month.
Julius - Heather's lawyer - and Paul's representative have so far declined to comment.
Last month, it was reported Heather and Paul - who were married for four years - had reached an out of court settlement worth $60 million.
However, legal firm Mishcon de Reya, who represent Heather, quickly issued a statement denying Paul had made an offer.
