The two sides spent much of Sunday negotiating and are aiming to wrap up the deal Monday, the source said.
Last week, Johnson's agent, C.J. LaBoy, said he was "very optimistic" after describing talks as being "productive," in an email to The Associated Press.
Johnson, 25, led the Bills with 76 catches, joining Eric Moulds as the only Bills receivers to have 75 in consecutive seasons.
He also became the first Bills receiver in the franchise's 52-year history to post back-to-back 1,000 yard seasons. He finished with 1,004 yards.
Johnson, however, was penalized twice in five weeks for excessive touchdown celebrations -- the second leading to a benching by coach Chan Gailey for the final three quarters of a season-ending 49-21 loss at New England.
Adam Schefter is ESPN's NFL Insider. Information from The Associated Press was used in this report.