Murray led Great Britain to their first Davis Cup title since 1936 over the weekend, while English underdog Fury recorded a stunning victory to end Wladimir Klitschko's reign as world champion in Dusseldorf on Saturday.
While Murray's success in the team tennis and Fury's triumph in the ring earned them late entries onto the shortlist, the favourite according to William Hill bookmakers is Jessica Ennis-Hill, who was crowned world heptathlon champion in August, 13 months after giving birth.
There are two other contenders who enjoyed track and field success at the World Championships in Beijing.
One is Mo Farah, who secured a unique 'treble double' in distance running having won golds in both 5,000 metres and 10,000m at two World Championships and the Olympics.
The other is Greg Rutherford, the long-jump world champion who became only the fifth British athlete to hold Olympic, Commonwealth, European and world titles at the same time.
Last year's winner of the BBC award, Lewis Hamilton, is also shortlisted again after successfully defending his Formula One world champion title, his third drivers' championship in all.
Tour de France winner Chris Froome is nominated, as is fellow cyclist Lizzie Armitstead, who won the UCI World Road Championship.
Lucy Bronze, the defender who helped England's women's football team to third place in the World Cup and was shortlisted for the player of the tournament, is also nominated.
Kevin Sinfield, who retired from rugby league after an extraordinary career, including winning the Challenge Cup, League Leaders' Shield and the Super League title in his final season, before switching codes to union is also included.
The final contenders are Adam Peaty, who became the first British swimmer to win three gold medals at a single World Championships, and Max Whitlock, who ended Britain's 112-year wait for a men's gold medallist at the World Gymnastics Championships.
The winner will be decided by a public vote and announced at a live event from Belfast on December 20.