Sir Paul is worth an estimated £20m more than last year and enjoys a significant boost from his American heiress wife's £150m stake in her family's US trucking business.
It puts him well ahead of his nearest rival on the list, Andrew Lloyd Webber, who is estimated to be worth £650m.
The full list will be published by the newspaper on 26 April.
Of the 1,000 richest people in the UK and the 250 wealthiest in Ireland, the list puts Irish band U2 at third place with £431m.
Pop veteran Sir Elton John and Rolling Stones' frontman Sir Mick Jagger follow with their fortunes, thought to be worth £270m and £225m respectively.
1. Sir Paul McCartney and Nancy Shevell £730m
2. Lord Lloyd-Webber £650m
3. U2 £431m
4. Sir Elton John £270m
5. Sir Mick Jagger £225m
6. Keith Richards £210m
7. Michael Flatley £195m
=8. Ringo Starr £180m
=8. Sting £180m
9. Roger Waters £160m
For the third year running, singer-songwriter Adele, who is 26, has topped the list for young music millionaires under the age of 30 in Britain and Ireland, with an estimated fortune of £50m.
The four members of the boy band One Direction plus former band mate Zayn Malik share the second spot with an estimated wealth of £25m each.
One of the fastest rising new young stars is the 22-year-old soul singer Sam Smith, who is currently in the running for the prize for the most played song of the past year at The Ivor Novello songwriting awards.
The Stay With Me singer has amassed a personal fortune of £12m since rising to fame in late 2012.
Former Beatle Sir Paul McCartney has topped the Sunday Times Rich List of musicians with his £730m fortune.