Local media have reported Frenchman Anelka's management is in talks with the Chinese Super League club to forgo the final year of his contract, while Ivory Coast striker Drogba has been linked with a move back to Europe.
Drogba signed a two-and-a-half year deal reportedly worth around $300,000 per week in June, but the Shanghai-based Oriental Sports Daily said last week the 34-year-old was owed wages in the wake of a long-running equity row at the club.
Drogba applied for permission last month to leave Shanghai on loan before the January transfer window, but FIFA refused the request.
He has since returned to Chelsea to train at his former club in a bid to stay sharp for the African Nations Cup, which starts on January 19, fuelling speculation of a return to Europe.
A Shenhua spokesman confirmed to Reuters that Anelka was in talks with the club regarding his playing future, but declined to comment on whether Drogba had confirmed he would return to Shanghai after playing the African Nations Cup.
"They are still contracted players, and nothing has changed in that regard," he told Reuters by telephone on Tuesday.
The loss of the two high-profile forwards would come as a huge blow to the club and the Chinese Super League, which had trumpeted their arrivals as a sign of faith in the competition after years of being mired in corruption scandals.
Despite the fanfare, Shanghai's hopes of a run to the title on the back of their prized recruits crumbled during a wayward season blighted by in-fighting.
Shanghai started the year favourites to win the 2012 premiership, thanks to a massive cash injection from their colourful owner Zhu Jun.
The online gaming tycoon secured former Arsenal, Liverpool and Chelsea striker Anelka's signature a year ago, and further raised expectations by appointing former Fulham manager Jean Tigana as head coach.
Shanghai's season started to unravel when Tigana was hit by a player revolt after just five games in charge.
After Tigana's entire backroom staff were fired, the Frenchman himself was sacked shortly after arriving at the stadium for Shanghai's match against Tianjin Teda.
Tigana left the stadium before kick-off, and Shanghai were left to play out a 1-0 loss to Tianjin with a virtually empty home bench.
Tigana's replacement raised further eyebrows, with Anelka, no stranger to clashes with authority, sensationally named player-coach.
The unorthodox appointment did little to improve Shenhua's fortunes, though Anelka threatened to quit when he learned of plans to replace him.
"If there is still no one to support me and (they) continue to play little tricks behind my back... then I will quickly decide whether or not to retire," he told Shanghai media.
However, Anelka was placated and former Argentina coach Sergio Batista arrived in May to become the team's third head coach in 2012.
Batista steadied the ship with some solid results, and Drogba's arrival in July to a hero's welcome from hundreds of fans at Shanghai's Pudong airport, briefly raised hopes the season could be saved.
Drogba proved an instant hit, scoring two goals on his home debut in a 5-1 drubbing of local rivals Hangzhou Greentown in early August, but the club was again in crisis a few weeks later due to a festering boardroom dispute.
Owner Zhu threatened to withhold player wages, claiming other shareholders had reneged on an agreement to give him a majority stake in the club.
While Drogba lived up to his promise with eight goals in 11 appearances, Anelka largely struggled amid the hype, finding the back of the net only three times in his 22 matches.
Shanghai was the ultimate loser, the club's ninth-placed finish in the 16-team competition proving that money alone rarely buys success.