The Daily Mirror and The Sun tabloids said on Saturday that McCartney and Shevell, who posted wedding banns last month, would be married at London's Marylebone Register Office before 30 invited guests.
It is the same venue where McCartney married his late first wife Linda in 1969, at the height of the Beatles' fame. Linda died of breast cancer in 1998.
They have been given special dispensation by the council to marry on Sunday and have chosen what would have been the groom's former bandmate John Lennon's 71st birthday for their big day, the Daily Mirror said.
McCartney's partnership with Lennon in The Beatles in 1960s Britain produced some of the most famous and enduring pop songs of the past 50 years.
The Daily Mirror quoted a source close to McCartney as saying: The wedding is in keeping with the way Paul and Nancy have conducted their entire relationship -- low-key, understated and fueled by goodwill.
Neither of them want a huge fuss made and the main priority for both was family.
The wedding will be followed by a reception held in McCartney's back garden, the paper said.
Shevell, 51, who is divorced, and McCartney, 69, are believed to have begun dating after his bitter split with second wife, former model Heather Mills.
McCartney and Mills were wed in 2002 in a lavish ceremony at Castle Leslie in Ireland estimated by celebrity magazines to have cost around $3.2 million.
But four years later they confirmed they had separated, blaming media intrusion into their private lives, and in 2008 the couple divorced.
McCartney has a son, two daughters and a stepdaughter from his marriage to Linda and a daughter from his union with Mills.